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# Elements of free modules

Elements of free modules

AUTHORS:

- William Stein
- Josh Kantor
- Thomas Feulner (2012-11): Added `FreeModuleElement.hamming_weight()` and `FreeModuleElement_generic_sparse.hamming_weight()`

TODO: Change to use a get_unsafe / set_unsafe, etc., structure exactly like with matrices, since we’ll have to define a bunch of special purpose implementations of vectors easily and systematically.

EXAMPLES: We create a vector space over \(\QQ\) and a subspace of this space.

    sage: V = QQ^5
    sage: W = V.span([V.1, V.2])

Arithmetic operations always return something in the ambient space, since there is a canonical map from \(W\) to \(V\) but not from \(V\) to \(W\).

    sage: parent(W.0 + V.1)
    Vector space of dimension 5 over Rational Field
    sage: parent(V.1 + W.0)
    Vector space of dimension 5 over Rational Field
    sage: W.0 + V.1
    (0, 2, 0, 0, 0)
    sage: W.0 - V.0
    (-1, 1, 0, 0, 0)

Next we define modules over \(\ZZ\) and a finite field.

    sage: K = ZZ^5
    sage: M = GF(7)^5

Arithmetic between the \(\QQ\) and \(\ZZ\) modules is defined, and the result is always over \(\QQ\), since there is a canonical coercion map to \(\QQ\).

    sage: K.0 + V.1
    (1, 1, 0, 0, 0)
    sage: parent(K.0 + V.1)
    Vector space of dimension 5 over Rational Field

Since there is no canonical coercion map to the finite field from \(\QQ\) the following arithmetic is not defined:

    sage: V.0 + M.0
    Traceback (most recent call last):
    ...
    TypeError: unsupported operand parent(s) for '+': 'Vector space of dimension 5 over Rational Field' and 'Vector space of dimension 5 over Finite Field of size 7'

However, there is a map from \(\ZZ\) to the finite field, so the following is defined, and the result is in the finite field.

    sage: w = K.0 + M.0; w
    (2, 0, 0, 0, 0)
    sage: parent(w)
    Vector space of dimension 5 over Finite Field of size 7
    sage: parent(M.0 + K.0)
    Vector space of dimension 5 over Finite Field of size 7

Matrix vector multiply:

    sage: MS = MatrixSpace(QQ,3)
    sage: A = MS([0,1,0,1,0,0,0,0,1])
    sage: V = QQ^3
    sage: v = V([1,2,3])
    sage: v * A
    (2, 1, 3)

TESTS:

    sage: D = 46341
    sage: u = 7
    sage: R = Integers(D)
    sage: p = matrix(R,[[84, 97, 55, 58, 51]])
    sage: 2*p.row(0)
    (168, 194, 110, 116, 102)

class `sage.modules.free_module_element.``FreeModuleElement`

  Bases: `sage.structure.element.Vector`

  An element of a generic free module.

  `Mod`(p)

    EXAMPLES:

        sage: V = vector(ZZ, [5, 9, 13, 15])
        sage: V.Mod(7)
        (5, 2, 6, 1)
        sage: parent(V.Mod(7))
        Vector space of dimension 4 over Ring of integers modulo 7

  `additive_order`()

    Return the additive order of self.

    EXAMPLES:

        sage: v = vector(Integers(4), [1,2])
        sage: v.additive_order()
        4

        sage: v = vector([1,2,3])
        sage: v.additive_order()
        +Infinity

        sage: v = vector(Integers(30), [6, 15]); v
        (6, 15)
        sage: v.additive_order()
        10
        sage: 10*v
        (0, 0)

  `apply_map`(phi, R=None, sparse=None)

    Apply the given map phi (an arbitrary Python function or callable object) to this free module element. If R is not given, automatically determine the base ring of the resulting element.

    INPUT:
      sparse – True or False will control whether the result
        is sparse. By default, the result is sparse iff self is sparse.
    - `phi` - arbitrary Python function or callable object
    - `R` - (optional) ring

    OUTPUT: a free module element over R

    EXAMPLES:

        sage: m = vector([1,x,sin(x+1)])
        sage: m.apply_map(lambda x: x^2)
        (1, x^2, sin(x + 1)^2)
        sage: m.apply_map(sin)
        (sin(1), sin(x), sin(sin(x + 1)))

        sage: m = vector(ZZ, 9, range(9))
        sage: k.<a> = GF(9)
        sage: m.apply_map(k)
        (0, 1, 2, 0, 1, 2, 0, 1, 2)

    In this example, we explicitly specify the codomain.

        sage: s = GF(3)
        sage: f = lambda x: s(x)
        sage: n = m.apply_map(f, k); n
        (0, 1, 2, 0, 1, 2, 0, 1, 2)
        sage: n.parent()
        Vector space of dimension 9 over Finite Field in a of size 3^2

    If your map sends 0 to a non-zero value, then your resulting vector is not mathematically sparse:

        sage: v = vector([0] * 6 + [1], sparse=True); v
        (0, 0, 0, 0, 0, 0, 1)
        sage: v2 = v.apply_map(lambda x: x+1); v2
        (1, 1, 1, 1, 1, 1, 2)

    but it’s still represented with a sparse data type:

        sage: parent(v2)
        Ambient sparse free module of rank 7 over the principal ideal domain Integer Ring

    This data type is inefficient for dense vectors, so you may want to specify sparse=False:

        sage: v2 = v.apply_map(lambda x: x+1, sparse=False); v2
        (1, 1, 1, 1, 1, 1, 2)
        sage: parent(v2)
        Ambient free module of rank 7 over the principal ideal domain Integer Ring

    Or if you have a map that will result in mostly zeroes, you may want to specify sparse=True:

        sage: v = vector(srange(10))
        sage: v2 = v.apply_map(lambda x: 0 if x else 1, sparse=True); v2
        (1, 0, 0, 0, 0, 0, 0, 0, 0, 0)
        sage: parent(v2)
        Ambient sparse free module of rank 10 over the principal ideal domain Integer Ring

    TESTS:

        sage: m = vector(SR,[])
        sage: m.apply_map(lambda x: x*x) == m
        True

    Check that we don’t unnecessarily apply phi to 0 in the sparse case:

        sage: m = vector(ZZ, range(1, 4), sparse=True)
        sage: m.apply_map(lambda x: 1/x)
        (1, 1/2, 1/3)

        sage: parent(vector(RDF, (), sparse=True).apply_map(lambda x: x, sparse=True))
        Sparse vector space of dimension 0 over Real Double Field
        sage: parent(vector(RDF, (), sparse=True).apply_map(lambda x: x, sparse=False))
        Vector space of dimension 0 over Real Double Field
        sage: parent(vector(RDF, (), sparse=False).apply_map(lambda x: x, sparse=True))
        Sparse vector space of dimension 0 over Real Double Field
        sage: parent(vector(RDF, (), sparse=False).apply_map(lambda x: x, sparse=False))
        Vector space of dimension 0 over Real Double Field

    Check that the bug in trac ticket #14558 has been fixed:

        sage: F.<a> = GF(9)
        sage: v = vector([a, 0,0,0], sparse=True)
        sage: f = F.hom([a**3])
        sage: v.apply_map(f)
        (2*a + 1, 0, 0, 0)

  `change_ring`(R)

    Change the base ring of this vector, by coercing each element of this vector into R.

    EXAMPLES:

        sage: v = vector(QQ['x,y'], [1..5]); v.change_ring(GF(3))
        (1, 2, 0, 1, 2)

  `column`()

    Return a matrix with a single column and the same entries as the vector `self`.

    OUTPUT:

    A matrix over the same ring as the vector (or free module element), with a single column. The entries of the column are identical to those of the vector, and in the same order.

    EXAMPLES:

        sage: v = vector(ZZ, [1,2,3])
        sage: w = v.column(); w
        [1]
        [2]
        [3]
        sage: w.parent()
        Full MatrixSpace of 3 by 1 dense matrices over Integer Ring

        sage: x = vector(FiniteField(13), [2,4,8,16])
        sage: x.column()
        [2]
        [4]
        [8]
        [3]

    There is more than one way to get one-column matrix from a vector. The `column` method is about equally efficient to making a row and then taking a transpose. Notice that supplying a vector to the matrix constructor demonstrates Sage’s preference for rows.

        sage: x = vector(RDF, [sin(i*pi/20) for i in range(10)])
        sage: x.column() == matrix(x).transpose()
        True
        sage: x.column() == x.row().transpose()
        True

    Sparse or dense implementations are preserved.

        sage: d = vector(RR, [1.0, 2.0, 3.0])
        sage: s = vector(CDF, {2:5.0+6.0*I})
        sage: dm = d.column()
        sage: sm = s.column()
        sage: all([d.is_dense(), dm.is_dense(), s.is_sparse(), sm.is_sparse()])
        True

    TESTS:

    The `column()` method will return a specified column of a matrix as a vector. So here are a couple of round-trips.

        sage: A = matrix(ZZ, [[1],[2],[3]])
        sage: A == A.column(0).column()
        True
        sage: v = vector(ZZ, [4,5,6])
        sage: v == v.column().column(0)
        True

    And a very small corner case.

        sage: v = vector(ZZ, [])
        sage: w = v.column()
        sage: w.parent()
        Full MatrixSpace of 0 by 1 dense matrices over Integer Ring

  `conjugate`()

    Returns a vector where every entry has been replaced by its complex conjugate.

    OUTPUT:

    A vector of the same length, over the same ring, but with each entry replaced by the complex conjugate, as implemented by the `conjugate()` method for elements of the base ring, which is presently always complex conjugation.

    EXAMPLES:

        sage: v = vector(CDF, [2.3 - 5.4*I, -1.7 + 3.6*I])
        sage: w = v.conjugate(); w
        (2.3 + 5.4*I, -1.7 - 3.6*I)
        sage: w.parent()
        Vector space of dimension 2 over Complex Double Field

    Even if conjugation seems nonsensical over a certain ring, this method for vectors cooperates silently.

        sage: u = vector(ZZ, range(6))
        sage: u.conjugate()
        (0, 1, 2, 3, 4, 5)

    Sage implements a few specialized subfields of the complex numbers, such as the cyclotomic fields. This example uses such a field containing a primitive 7-th root of unity named `a`.

        sage: F.<a> = CyclotomicField(7)
        sage: v = vector(F, [a^i for i in range(7)])
        sage: v
        (1, a, a^2, a^3, a^4, a^5, -a^5 - a^4 - a^3 - a^2 - a - 1)
        sage: v.conjugate()
        (1, -a^5 - a^4 - a^3 - a^2 - a - 1, a^5, a^4, a^3, a^2, a)

    Sparse vectors are returned as such.

        sage: v = vector(CC, {1: 5 - 6*I, 3: -7*I}); v
        (0.000000000000000, 5.00000000000000 - 6.00000000000000*I, 0.000000000000000, -7.00000000000000*I)
        sage: v.is_sparse()
        True
        sage: vc = v.conjugate(); vc
        (0.000000000000000, 5.00000000000000 + 6.00000000000000*I, 0.000000000000000, 7.00000000000000*I)
        sage: vc.conjugate()
        (0.000000000000000, 5.00000000000000 - 6.00000000000000*I, 0.000000000000000, -7.00000000000000*I)

    TESTS:

        sage: n = 15
        sage: x = vector(CDF, [sin(i*pi/n)+cos(i*pi/n)*I for i in range(n)])
        sage: x + x.conjugate() in RDF^n
        True
        sage: I*(x - x.conjugate()) in RDF^n
        True

    The parent of the conjugate is the same as that of the original vector. We test this by building a specialized vector space with a non-standard inner product, and constructing a test vector in this space.

        sage: V = VectorSpace(CDF, 2, inner_product_matrix = [[2,1],[1,5]])
        sage: v = vector(CDF, [2-3*I, 4+5*I])
        sage: w = V(v)
        sage: w.parent()
        Ambient quadratic space of dimension 2 over Complex Double Field
        Inner product matrix:
        [2.0 1.0]
        [1.0 5.0]
        sage: w.conjugate().parent()
        Ambient quadratic space of dimension 2 over Complex Double Field
        Inner product matrix:
        [2.0 1.0]
        [1.0 5.0]

  `cross_product`(right)

    Return the cross product of self and right, which is only defined for vectors of length 3 or 7.

    INPUT:

    - `right` - A vector of the same size as `self`, either degree three or degree seven.

    OUTPUT:

    The cross product (vector product) of `self` and `right`, a vector of the same size of `self` and `right`.

    This product is performed under the assumption that the basis vectors are orthonormal.

    EXAMPLES:

        sage: v = vector([1,2,3]); w = vector([0,5,-9])
        sage: v.cross_product(v)
        (0, 0, 0)
        sage: u = v.cross_product(w); u
        (-33, 9, 5)
        sage: u.dot_product(v)
        0
        sage: u.dot_product(w)
        0

    The cross product is defined for degree seven vectors as well. [WIKIPEDIA:CROSSPRODUCT] The 3-D cross product is achieved using the quaternians, whereas the 7-D cross product is achieved using the octions.

        sage: u = vector(QQ, [1, -1/3, 57, -9, 56/4, -4,1])
        sage: v = vector(QQ, [37, 55, -99/57, 9, -12, 11/3, 4/98])
        sage: u.cross_product(v)
        (1394815/2793, -2808401/2793, 39492/49, -48737/399, -9151880/2793, 62513/2793, -326603/171)

    The degree seven cross product is anticommutative.

        sage: u.cross_product(v) + v.cross_product(u)
        (0, 0, 0, 0, 0, 0, 0)

    The degree seven cross product is distributive across addition.

        sage: v = vector([-12, -8/9, 42, 89, -37, 60/99, 73])
        sage: u = vector([31, -42/7, 97, 80, 30/55, -32, 64])
        sage: w = vector([-25/4, 40, -89, -91, -72/7, 79, 58])
        sage: v.cross_product(u + w) - (v.cross_product(u) + v.cross_product(w))
        (0, 0, 0, 0, 0, 0, 0)

    The degree seven cross product respects scalar multiplication.

        sage: v = vector([2, 17, -11/5, 21, -6, 2/17, 16])
        sage: u = vector([-8, 9, -21, -6, -5/3, 12, 99])
        sage: (5*v).cross_product(u) - 5*(v.cross_product(u))
        (0, 0, 0, 0, 0, 0, 0)
        sage: v.cross_product(5*u) - 5*(v.cross_product(u))
        (0, 0, 0, 0, 0, 0, 0)
        sage: (5*v).cross_product(u) - (v.cross_product(5*u))
        (0, 0, 0, 0, 0, 0, 0)

    The degree seven cross product respects the scalar triple product.

        sage: v = vector([2,6,-7/4,-9/12,-7,12,9])
        sage: u = vector([22,-7,-9/11,12,15,15/7,11])
        sage: w = vector([-11,17,19,-12/5,44,21/56,-8])
        sage: v.dot_product(u.cross_product(w)) - w.dot_product(v.cross_product(u))
        0

    TESTS:

    Both vectors need to be of length three or both vectors need to be of length seven.

        sage: u = vector(range(7))
        sage: v = vector(range(3))
        sage: u.cross_product(v)
        Traceback (most recent call last):
        ...
        ArithmeticError: Cross product only defined for vectors of length three or seven, not (7 and 3)

    REFERENCES:

    [WIKIPEDIA:CROSSPRODUCT] | Algebraic Properties of the Cross Product http://en.wikipedia.org/wiki/Cross_product

    AUTHOR:

    Billy Wonderly (2010-05-11), Added 7-D Cross Product

  `degree`()

    Return the degree of this vector, which is simply the number of entries.

    EXAMPLES:

        sage: sage.modules.free_module_element.FreeModuleElement(QQ^389).degree()
        389
        sage: vector([1,2/3,8]).degree()
        3

  `denominator`()

    Return the least common multiple of the denominators of the entries of self.

    EXAMPLES:

        sage: v = vector([1/2,2/5,3/14])
        sage: v.denominator()
        70
        sage: 2*5*7
        70

    TESTS:

    The following was fixed in trac ticket #8800:

        sage: M = GF(5)^3
        sage: v = M((4,0,2))
        sage: v.denominator()
        1

  `dense_vector`()

    Return dense version of self. If self is dense, just return self; otherwise, create and return correspond dense vector.

    EXAMPLES:

        sage: vector([-1,0,3,0,0,0]).dense_vector().is_dense()
        True
        sage: vector([-1,0,3,0,0,0],sparse=True).dense_vector().is_dense()
        True
        sage: vector([-1,0,3,0,0,0],sparse=True).dense_vector()
        (-1, 0, 3, 0, 0, 0)

  `derivative`(*args)

    Derivative with respect to variables supplied in args.

    Multiple variables and iteration counts may be supplied; see documentation for the global derivative() function for more details.

    `diff()` is an alias of this function.

    EXAMPLES:

        sage: v = vector([1,x,x^2])
        sage: v.derivative(x)
        (0, 1, 2*x)
        sage: type(v.derivative(x)) == type(v)
        True
        sage: v = vector([1,x,x^2], sparse=True)
        sage: v.derivative(x)
        (0, 1, 2*x)
        sage: type(v.derivative(x)) == type(v)
        True
        sage: v.derivative(x,x)
        (0, 0, 2)

  `dict`(copy=True)

    Return dictionary of nonzero entries of self.

    INPUT:

    > - `copy` – bool (default: True)

    OUTPUT:

    > - Python dictionary

    EXAMPLES:

        sage: v = vector([0,0,0,0,1/2,0,3/14])
        sage: v.dict()
        {4: 1/2, 6: 3/14}

    In some cases when copy=False, we get back a dangerous reference:

        sage: v = vector({0:5, 2:3/7}, sparse=True)
        sage: v.dict(copy=False)
        {0: 5, 2: 3/7}
        sage: v.dict(copy=False)[0] = 18
        sage: v
        (18, 0, 3/7)

  `diff`(*args)

    Derivative with respect to variables supplied in args.

    Multiple variables and iteration counts may be supplied; see documentation for the global derivative() function for more details.

    `diff()` is an alias of this function.

    EXAMPLES:

        sage: v = vector([1,x,x^2])
        sage: v.derivative(x)
        (0, 1, 2*x)
        sage: type(v.derivative(x)) == type(v)
        True
        sage: v = vector([1,x,x^2], sparse=True)
        sage: v.derivative(x)
        (0, 1, 2*x)
        sage: type(v.derivative(x)) == type(v)
        True
        sage: v.derivative(x,x)
        (0, 0, 2)

  `dot_product`(right)

    Return the dot product of `self` and `right`, which is the sum of the product of the corresponding entries.

    INPUT:

    - `right` - a vector of the same degree as `self`. It does not need to belong to the same parent as `self`, so long as the necessary products and sums are defined.

    OUTPUT:

    If `self` and `right` are the vectors \(\vec{x}\) and \(\vec{y}\), of degree \(n\), then this method returns

    \[\sum_{i=1}^{n}x_iy_i\]

    Note

    The `inner_product()` is a more general version of this method, and the `hermitian_inner_product()` method may be more appropriate if your vectors have complex entries.

    EXAMPLES:

        sage: V = FreeModule(ZZ, 3)
        sage: v = V([1,2,3])
        sage: w = V([4,5,6])
        sage: v.dot_product(w)
        32

    The vectors may be from different vector spaces, provided the necessary operations make sense. Notice that coercion will generate a result of the same type, even if the order of the arguments is reversed.:

        sage: v = vector(ZZ, [1,2,3])
        sage: w = vector(FiniteField(3), [0,1,2])
        sage: ip = w.dot_product(v); ip
        2
        sage: ip.parent()
        Finite Field of size 3

        sage: ip = v.dot_product(w); ip
        2
        sage: ip.parent()
        Finite Field of size 3

    The dot product of a vector with itself is the 2-norm, squared.

        sage: v = vector(QQ, [3, 4, 7])
        sage: v.dot_product(v) - v.norm()^2
        0

    TESTS:

    The second argument must be a free module element.

        sage: v = vector(QQ, [1,2])
        sage: v.dot_product('junk')
        Traceback (most recent call last):
        ...
        TypeError: right must be a free module element

    The degrees of the arguments must match.

        sage: v = vector(QQ, [1,2])
        sage: w = vector(QQ, [1,2,3])
        sage: v.dot_product(w)
        Traceback (most recent call last):
        ...
        ArithmeticError: degrees (2 and 3) must be the same

    Check that vectors with different base rings play out nicely (trac ticket #3103):

        sage: vector(CDF, [2, 2]) * vector(ZZ, [1, 3])
        8.0

  `element`()

    Simply returns self. This is useful, since for many objects, self.element() returns a vector corresponding to self.

    EXAMPLES:

        sage: v = vector([1/2,2/5,0]); v
        (1/2, 2/5, 0)
        sage: v.element()
        (1/2, 2/5, 0)

  `get`(i)

    The get method is in some cases more efficient (and more dangerous) than __getitem__, because it is not guaranteed to do any error checking.

    EXAMPLES:

        sage: vector([1/2,2/5,0]).get(0)
        1/2
        sage: vector([1/2,2/5,0]).get(3)
        Traceback (most recent call last):
        ...
        IndexError: index out of range

  `hamming_weight`()

    Return the number of positions `i` such that `self[i] != 0`.

    EXAMPLES:

        sage: vector([-1,0,3,0,0,0,0.01]).hamming_weight()
        3

  `hermitian_inner_product`(right)

    Returns the dot product, but with the entries of the first vector conjugated beforehand.

    INPUT:

    - `right` - a vector of the same degree as `self`

    OUTPUT:

    If `self` and `right` are the vectors \(\vec{x}\) and \(\vec{y}\) of degree \(n\) then this routine computes

    \[\sum_{i=1}^{n}\overline{x}_i{y}_i\]

    where the bar indicates complex conjugation.

    Note

    If your vectors do not contain complex entries, then `dot_product()` will return the same result without the overhead of conjugating elements of `self`.

    If you are not computing a weighted inner product, and your vectors do not have complex entries, then the `dot_product()` will return the same result.

    EXAMPLES:

        sage: v = vector(CDF, [2+3*I, 5-4*I])
        sage: w = vector(CDF, [6-4*I, 2+3*I])
        sage: v.hermitian_inner_product(w)
        -2.0 - 3.0*I

    Sage implements a few specialized fields over the complex numbers, such as cyclotomic fields and quadratic number fields. So long as the base rings have a conjugate method, then the Hermitian inner product will be available.

        sage: Q.<a> = QuadraticField(-7)
        sage: a^2
        -7
        sage: v = vector(Q, [3+a, 5-2*a])
        sage: w = vector(Q, [6, 4+3*a])
        sage: v.hermitian_inner_product(w)
        17*a - 4

    The Hermitian inner product should be additive in each argument (we only need to test one), linear in each argument (with conjugation on the first scalar), and anti-commutative.

        sage: alpha = CDF(5.0 + 3.0*I)
        sage: u = vector(CDF, [2+4*I, -3+5*I, 2-7*I])
        sage: v = vector(CDF, [-1+3*I, 5+4*I, 9-2*I])
        sage: w = vector(CDF, [8+3*I, -4+7*I, 3-6*I])
        sage: (u+v).hermitian_inner_product(w) == u.hermitian_inner_product(w) + v.hermitian_inner_product(w)
        True
        sage: (alpha*u).hermitian_inner_product(w) == alpha.conjugate()*u.hermitian_inner_product(w)
        True
        sage: u.hermitian_inner_product(alpha*w) == alpha*u.hermitian_inner_product(w)
        True
        sage: u.hermitian_inner_product(v) == v.hermitian_inner_product(u).conjugate()
        True

    For vectors with complex entries, the Hermitian inner product has a more natural relationship with the 2-norm (which is the default for the `norm()` method). The norm squared equals the Hermitian inner product of the vector with itself.

        sage: v = vector(CDF, [-0.66+0.47*I, -0.60+0.91*I, -0.62-0.87*I, 0.53+0.32*I])
        sage: abs(v.norm()^2 - v.hermitian_inner_product(v)) < 1.0e-10
        True

    TESTS:

    This method is built on the `dot_product()` method, which allows for a wide variety of inputs. Any error handling happens there.

        sage: v = vector(CDF, [2+3*I])
        sage: w = vector(CDF, [5+2*I, 3+9*I])
        sage: v.hermitian_inner_product(w)
        Traceback (most recent call last):
        ...
        ArithmeticError: degrees (1 and 2) must be the same

  `inner_product`(right)

    Returns the inner product of `self` and `right`, possibly using an inner product matrix from the parent of `self`.

    INPUT:

    - `right` - a vector of the same degree as `self`

    OUTPUT:

    If the parent vector space does not have an inner product matrix defined, then this is the usual dot product (`dot_product()`). If `self` and `right` are considered as single column matrices, \(\vec{x}\) and \(\vec{y}\), and \(A\) is the inner product matrix, then this method computes

    \[\left(\vec{x}\right)^tA\vec{y}\]

    where \(t\) indicates the transpose.

    Note

    If your vectors have complex entries, the `hermitian_inner_product()` may be more appropriate for your purposes.

    EXAMPLES:

        sage: v = vector(QQ, [1,2,3])
        sage: w = vector(QQ, [-1,2,-3])
        sage: v.inner_product(w)
        -6
        sage: v.inner_product(w) == v.dot_product(w)
        True

    The vector space or free module that is the parent to `self` can have an inner product matrix defined, which will be used by this method. This matrix will be passed through to subspaces.

        sage: ipm = matrix(ZZ,[[2,0,-1], [0,2,0], [-1,0,6]])
        sage: M = FreeModule(ZZ, 3, inner_product_matrix = ipm)
        sage: v = M([1,0,0])
        sage: v.inner_product(v)
        2
        sage: K = M.span_of_basis([[0/2,-1/2,-1/2], [0,1/2,-1/2],[2,0,0]])
        sage: (K.0).inner_product(K.0)
        2
        sage: w = M([1,3,-1])
        sage: v = M([2,-4,5])
        sage: w.row()*ipm*v.column() == w.inner_product(v)
        True

    Note that the inner product matrix comes from the parent of `self`. So if a vector is not an element of the correct parent, the result could be a source of confusion.

        sage: V = VectorSpace(QQ, 2, inner_product_matrix=[[1,2],[2,1]])
        sage: v = V([12, -10])
        sage: w = vector(QQ, [10,12])
        sage: v.inner_product(w)
        88
        sage: w.inner_product(v)
        0
        sage: w = V(w)
        sage: w.inner_product(v)
        88

    Note

    The use of an inner product matrix makes no restrictions on the nature of the matrix. In particular, in this context it need not be Hermitian and positive-definite (as it is in the example above).

    TESTS:

    Most error handling occurs in the `dot_product()` method. But with an inner product defined, this method will check that the input is a vector or free module element.

        sage: W = VectorSpace(RDF, 2, inner_product_matrix = matrix(RDF, 2, [1.0,2.0,3.0,4.0]))
        sage: v = W([2.0, 4.0])
        sage: v.inner_product(5)
        Traceback (most recent call last):
        ...
        TypeError: right must be a free module element

  `integral`(*args, **kwds)

    Returns a symbolic integral of the vector, component-wise.

    `integrate()` is an alias of the function.

    EXAMPLES:

        sage: t=var('t')
        sage: r=vector([t,t^2,sin(t)])
        sage: r.integral(t)
        (1/2*t^2, 1/3*t^3, -cos(t))
        sage: integrate(r,t)
        (1/2*t^2, 1/3*t^3, -cos(t))
        sage: r.integrate(t,0,1)
        (1/2, 1/3, -cos(1) + 1)

  `integrate`(*args, **kwds)

    Returns a symbolic integral of the vector, component-wise.

    `integrate()` is an alias of the function.

    EXAMPLES:

        sage: t=var('t')
        sage: r=vector([t,t^2,sin(t)])
        sage: r.integral(t)
        (1/2*t^2, 1/3*t^3, -cos(t))
        sage: integrate(r,t)
        (1/2*t^2, 1/3*t^3, -cos(t))
        sage: r.integrate(t,0,1)
        (1/2, 1/3, -cos(1) + 1)

  `is_dense`()

    Return True if this is a dense vector, which is just a statement about the data structure, not the number of nonzero entries.

    EXAMPLES:

        sage: vector([1/2,2/5,0]).is_dense()
        True
        sage: vector([1/2,2/5,0],sparse=True).is_dense()
        False

  `is_immutable`()

    Return True if this vector is immutable, i.e., the entries cannot be changed.

    EXAMPLES:

        sage: v = vector(QQ['x,y'], [1..5]); v.is_immutable()
        False
        sage: v.set_immutable()
        sage: v.is_immutable()
        True

  `is_mutable`()

    Return True if this vector is mutable, i.e., the entries can be changed.

    EXAMPLES:

        sage: v = vector(QQ['x,y'], [1..5]); v.is_mutable()
        True
        sage: v.set_immutable()
        sage: v.is_mutable()
        False

  `is_sparse`()

    Return True if this is a sparse vector, which is just a statement about the data structure, not the number of nonzero entries.

    EXAMPLES:

        sage: vector([1/2,2/5,0]).is_sparse()
        False
        sage: vector([1/2,2/5,0],sparse=True).is_sparse()
        True

  `is_vector`()

    Return True, since this is a vector.

    EXAMPLES:

        sage: vector([1/2,2/5,0]).is_vector()
        True

  `iteritems`()

    Return iterator over self.

    EXAMPLES:

        sage: v = vector([1,2/3,pi])
        sage: v.iteritems()
        <dictionary-itemiterator object at ...>
        sage: list(v.iteritems())
        [(0, 1), (1, 2/3), (2, pi)]

  `lift`()

    EXAMPLES:

        sage: V = vector(Integers(7), [5, 9, 13, 15]) ; V
        (5, 2, 6, 1)
        sage: V.lift()
        (5, 2, 6, 1)
        sage: parent(V.lift())
        Ambient free module of rank 4 over the principal ideal domain Integer Ring

  `list`(copy=True)

    Return list of elements of self.

    INPUT:

    > - copy – bool, whether returned list is a copy that is safe to change, is ignored.

    EXAMPLES:

        sage: P.<x,y,z> = QQ[]
        sage: v = vector([x,y,z], sparse=True)
        sage: type(v)
        <type 'sage.modules.free_module_element.FreeModuleElement_generic_sparse'>
        sage: a = v.list(); a
        [x, y, z]
        sage: a[0] = x*y; v
        (x, y, z)

    The optional argument `copy` is ignored:

        sage: a = v.list(copy=False); a
        [x, y, z]
        sage: a[0] = x*y; v
        (x, y, z)

  `list_from_positions`(positions)

    Return list of elements chosen from this vector using the given positions of this vector.

    INPUT:

    > - positions – iterable of ints

    EXAMPLES:

        sage: v = vector([1,2/3,pi])
        sage: v.list_from_positions([0,0,0,2,1])
        [1, 1, 1, pi, 2/3]

  `monic`()

    Return this vector divided through by the first nonzero entry of this vector.

    EXAMPLES:

        sage: v = vector(QQ, [0, 4/3, 5, 1, 2])
        sage: v.monic()
        (0, 1, 15/4, 3/4, 3/2)

  `nintegral`(*args, **kwds)

    Returns a numeric integral of the vector, component-wise, and the result of the nintegral command on each component of the input.

    `nintegrate()` is an alias of the function.

    EXAMPLES:

        sage: t=var('t')
        sage: r=vector([t,t^2,sin(t)])
        sage: vec,answers=r.nintegral(t,0,1)
        sage: vec
        (0.5, 0.3333333333333334, 0.4596976941318602)
        sage: type(vec)
        <type 'sage.modules.vector_real_double_dense.Vector_real_double_dense'>
        sage: answers
        [(0.5, 5.551115123125784e-15, 21, 0), (0.3333333333333..., 3.70074341541719e-15, 21, 0), (0.45969769413186..., 5.103669643922841e-15, 21, 0)]

        sage: r=vector([t,0,1], sparse=True)
        sage: r.nintegral(t,0,1)
        ((0.5, 0.0, 1.0), {0: (0.5, 5.551115123125784e-15, 21, 0), 2: (1.0, 1.11022302462515...e-14, 21, 0)})

  `nintegrate`(*args, **kwds)

    Returns a numeric integral of the vector, component-wise, and the result of the nintegral command on each component of the input.

    `nintegrate()` is an alias of the function.

    EXAMPLES:

        sage: t=var('t')
        sage: r=vector([t,t^2,sin(t)])
        sage: vec,answers=r.nintegral(t,0,1)
        sage: vec
        (0.5, 0.3333333333333334, 0.4596976941318602)
        sage: type(vec)
        <type 'sage.modules.vector_real_double_dense.Vector_real_double_dense'>
        sage: answers
        [(0.5, 5.551115123125784e-15, 21, 0), (0.3333333333333..., 3.70074341541719e-15, 21, 0), (0.45969769413186..., 5.103669643922841e-15, 21, 0)]

        sage: r=vector([t,0,1], sparse=True)
        sage: r.nintegral(t,0,1)
        ((0.5, 0.0, 1.0), {0: (0.5, 5.551115123125784e-15, 21, 0), 2: (1.0, 1.11022302462515...e-14, 21, 0)})

  `nonzero_positions`()

    Return the sorted list of integers `i` such that `self[i] != 0`.

    EXAMPLES:

        sage: vector([-1,0,3,0,0,0,0.01]).nonzero_positions()
        [0, 2, 6]

  `norm`(p=None)

    Return the \(p\)-norm of `self`.

    INPUT:

    - `p` - default: 2 - `p` can be a real number greater than 1, infinity (`oo` or `Infinity`), or a symbolic expression.

      > - \(p=1\): the taxicab (Manhattan) norm
      > - \(p=2\): the usual Euclidean norm (the default)
      > - \(p=\infty\): the maximum entry (in absolute value)

    Note

    See also `sage.misc.functional.norm()`

    EXAMPLES:

        sage: v = vector([1,2,-3])
        sage: v.norm(5)
        276^(1/5)

    The default is the usual Euclidean norm.

        sage: v.norm()
        sqrt(14)
        sage: v.norm(2)
        sqrt(14)

    The infinity norm is the maximum size (in absolute value) of the entries.

        sage: v.norm(Infinity)
        3
        sage: v.norm(oo)
        3

    Real or symbolic values may be used for `p`.

        sage: v=vector(RDF,[1,2,3])
        sage: v.norm(5)
        3.077384885394063
        sage: v.norm(pi/2)
        4.216595864704748
        sage: _=var('a b c d p'); v=vector([a, b, c, d])
        sage: v.norm(p)
        (abs(a)^p + abs(b)^p + abs(c)^p + abs(d)^p)^(1/p)

    Notice that the result may be a symbolic expression, owing to the necessity of taking a square root (in the default case). These results can be converted to numerical values if needed.

        sage: v = vector(ZZ, [3,4])
        sage: nrm = v.norm(); nrm
        5
        sage: nrm.parent()
        Rational Field

        sage: v = vector(QQ, [3, 5])
        sage: nrm = v.norm(); nrm
        sqrt(34)
        sage: nrm.parent()
        Symbolic Ring
        sage: numeric = N(nrm); numeric
        5.83095189484...
        sage: numeric.parent()
        Real Field with 53 bits of precision

    TESTS:

    The value of `p` must be greater than, or equal to, one.

        sage: v = vector(QQ, [1,2])
        sage: v.norm(0.99)
        Traceback (most recent call last):
        ...
        ValueError: 0.990000 is not greater than or equal to 1

    Norm works with python integers (see trac ticket #13502).

        sage: v = vector(QQ, [1,2])
        sage: v.norm(int(2))
        sqrt(5)

  `normalize`()

    This function is deprecated. For division by the p-norm use ‘normalized’, and for division by the first nonzero entry use ‘monic’ (previously the purpose of this function).

    EXAMPLES:

        sage: v = vector(QQ, [0, 4/3, 5, 1, 2])
        sage: v.normalize()
        doctest:...: DeprecationWarning: 'normalize' is deprecated...
        (0, 1, 15/4, 3/4, 3/2)

  `normalized`(p=None)

    Return the input vector divided by the p-norm.

    INPUT:

    - “p” - default: 2 - p value for the norm

    EXAMPLES:

        sage: v = vector(QQ, [4, 1, 3, 2])
        sage: v.normalized()
        (2/15*sqrt(30), 1/30*sqrt(30), 1/10*sqrt(30), 1/15*sqrt(30))
        sage: sum(v.normalized(1))
        1

    Note that normalizing the vector may change the base ring:

        sage: v.base_ring() == v.normalized().base_ring()
        False
        sage: u = vector(RDF, [-3, 4, 6, 9])
        sage: u.base_ring() == u.normalized().base_ring()
        True

  `outer_product`(right)

    Returns a matrix, the outer product of two vectors `self` and `right`.

    INPUT:

    - `right` - a vector (or free module element) of any size, whose elements are compatible (with regard to multiplication) with the elements of `self`.

    OUTPUT:

    The outer product of two vectors \(x\) and \(y\) (respectively `self` and `right`) can be described several ways. If we interpret \(x\) as a \(m\times 1\) matrix and interpret \(y\) as a \(1\times n\) matrix, then the outer product is the \(m\times n\) matrix from the usual matrix product \(xy\). Notice how this is the “opposite” in some ways from an inner product (which would require \(m=n\)).

    If we just consider vectors, use each entry of \(x\) to create a scalar multiples of the vector \(y\) and use these vectors as the rows of a matrix. Or use each entry of \(y\) to create a scalar multiples of \(x\) and use these vectors as the columns of a matrix.

    EXAMPLES:

        sage: u = vector(QQ, [1/2, 1/3, 1/4, 1/5])
        sage: v = vector(ZZ, [60, 180, 600])
        sage: u.outer_product(v)
        [ 30  90 300]
        [ 20  60 200]
        [ 15  45 150]
        [ 12  36 120]
        sage: M = v.outer_product(u); M
        [ 30  20  15  12]
        [ 90  60  45  36]
        [300 200 150 120]
        sage: M.parent()
        Full MatrixSpace of 3 by 4 dense matrices over Rational Field

    The more general `sage.matrix.matrix2.tensor_product()` is an operation on a pair of matrices. If we construe a pair of vectors as a column vector and a row vector, then an outer product and a tensor product are identical. Thus \(tensor_product\) is a synonym for this method.

        sage: u = vector(QQ, [1/2, 1/3, 1/4, 1/5])
        sage: v = vector(ZZ, [60, 180, 600])
        sage: u.tensor_product(v) == (u.column()).tensor_product(v.row())
        True

    The result is always a dense matrix, no matter if the two vectors are, or are not, dense.

        sage: d = vector(ZZ,[4,5], sparse=False)
        sage: s = vector(ZZ, [1,2,3], sparse=True)
        sage: dd = d.outer_product(d)
        sage: ds = d.outer_product(s)
        sage: sd = s.outer_product(d)
        sage: ss = s.outer_product(s)
        sage: all([dd.is_dense(), ds.is_dense(), sd.is_dense(), dd.is_dense()])
        True

    Vectors with no entries do the right thing.

        sage: v = vector(ZZ, [])
        sage: z = v.outer_product(v)
        sage: z.parent()
        Full MatrixSpace of 0 by 0 dense matrices over Integer Ring

    There is a fair amount of latitude in the value of the `right` vector, and the matrix that results can have entries from a new ring large enough to contain the result. If you know better, you can sometimes bring the result down to a less general ring.

        sage: R.<t> = ZZ[]
        sage: v = vector(R, [12, 24*t])
        sage: w = vector(QQ, [1/2, 1/3, 1/4])
        sage: op = v.outer_product(w)
        sage: op
        [   6    4    3]
        [12*t  8*t  6*t]
        sage: op.base_ring()
        Univariate Polynomial Ring in t over Rational Field
        sage: m = op.change_ring(R); m
        [   6    4    3]
        [12*t  8*t  6*t]
        sage: m.base_ring()
        Univariate Polynomial Ring in t over Integer Ring

    But some inputs are not compatible, even if vectors.

        sage: w = vector(GF(5), [1,2])
        sage: v = vector(GF(7), [1,2,3,4])
        sage: z = w.outer_product(v)
        Traceback (most recent call last):
        ...
        TypeError: unsupported operand parent(s) for '*': 'Full MatrixSpace of 2 by 1 dense matrices over Finite Field of size 5' and 'Full MatrixSpace of 1 by 4 dense matrices over Finite Field of size 7'

    And some inputs don’t make any sense at all.

        sage: w=vector(QQ, [5,10])
        sage: z=w.outer_product(6)
        Traceback (most recent call last):
        ...
        TypeError: right operand in an outer product must be a vector, not an element of Integer Ring

  `pairwise_product`(right)

    Return the pairwise product of self and right, which is a vector of the products of the corresponding entries.

    INPUT:

    - `right` - vector of the same degree as self. It need not be in the same vector space as self, as long as the coefficients can be multiplied.

    EXAMPLES:

        sage: V = FreeModule(ZZ, 3)
        sage: v = V([1,2,3])
        sage: w = V([4,5,6])
        sage: v.pairwise_product(w)
        (4, 10, 18)
        sage: sum(v.pairwise_product(w)) == v.dot_product(w)
        True

        sage: W = VectorSpace(GF(3),3)
        sage: w = W([0,1,2])
        sage: w.pairwise_product(v)
        (0, 2, 0)
        sage: w.pairwise_product(v).parent()
        Vector space of dimension 3 over Finite Field of size 3

    Implicit coercion is well defined (regardless of order), so we get 2 even if we do the dot product in the other order.

        sage: v.pairwise_product(w).parent()
        Vector space of dimension 3 over Finite Field of size 3

    TESTS:

        sage: x, y = var('x, y')

        sage: parent(vector(ZZ,[1,2]).pairwise_product(vector(ZZ,[1,2])))
        Ambient free module of rank 2 over the principal ideal domain Integer Ring
        sage: parent(vector(ZZ,[1,2]).pairwise_product(vector(QQ,[1,2])))
        Vector space of dimension 2 over Rational Field
        sage: parent(vector(QQ,[1,2]).pairwise_product(vector(ZZ,[1,2])))
        Vector space of dimension 2 over Rational Field
        sage: parent(vector(QQ,[1,2]).pairwise_product(vector(QQ,[1,2])))
        Vector space of dimension 2 over Rational Field

        sage: parent(vector(QQ,[1,2,3,4]).pairwise_product(vector(ZZ[x],[1,2,3,4])))
        Ambient free module of rank 4 over the principal ideal domain Univariate Polynomial Ring in x over Rational Field
        sage: parent(vector(ZZ[x],[1,2,3,4]).pairwise_product(vector(QQ,[1,2,3,4])))
        Ambient free module of rank 4 over the principal ideal domain Univariate Polynomial Ring in x over Rational Field

        sage: parent(vector(QQ,[1,2,3,4]).pairwise_product(vector(ZZ[x][y],[1,2,3,4])))
        Ambient free module of rank 4 over the integral domain Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
        sage: parent(vector(ZZ[x][y],[1,2,3,4]).pairwise_product(vector(QQ,[1,2,3,4])))
        Ambient free module of rank 4 over the integral domain Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field

        sage: parent(vector(QQ[x],[1,2,3,4]).pairwise_product(vector(ZZ[x][y],[1,2,3,4])))
        Ambient free module of rank 4 over the integral domain Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
        sage: parent(vector(ZZ[x][y],[1,2,3,4]).pairwise_product(vector(QQ[x],[1,2,3,4])))
        Ambient free module of rank 4 over the integral domain Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field

        sage: parent(vector(QQ[y],[1,2,3,4]).pairwise_product(vector(ZZ[x][y],[1,2,3,4])))
        Ambient free module of rank 4 over the integral domain Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
        sage: parent(vector(ZZ[x][y],[1,2,3,4]).pairwise_product(vector(QQ[y],[1,2,3,4])))
        Ambient free module of rank 4 over the integral domain Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field

        sage: parent(vector(ZZ[x],[1,2,3,4]).pairwise_product(vector(ZZ[y],[1,2,3,4])))
        Traceback (most recent call last):
        ...
        TypeError: no common canonical parent for objects with parents: 'Ambient free module of rank 4 over the integral domain Univariate Polynomial Ring in x over Integer Ring' and 'Ambient free module of rank 4 over the integral domain Univariate Polynomial Ring in y over Integer Ring'
        sage: parent(vector(ZZ[x],[1,2,3,4]).pairwise_product(vector(QQ[y],[1,2,3,4])))
        Traceback (most recent call last):
        ...
        TypeError: no common canonical parent for objects with parents: 'Ambient free module of rank 4 over the integral domain Univariate Polynomial Ring in x over Integer Ring' and 'Ambient free module of rank 4 over the principal ideal domain Univariate Polynomial Ring in y over Rational Field'
        sage: parent(vector(QQ[x],[1,2,3,4]).pairwise_product(vector(ZZ[y],[1,2,3,4])))
        Traceback (most recent call last):
        ...
        TypeError: no common canonical parent for objects with parents: 'Ambient free module of rank 4 over the principal ideal domain Univariate Polynomial Ring in x over Rational Field' and 'Ambient free module of rank 4 over the integral domain Univariate Polynomial Ring in y over Integer Ring'
        sage: parent(vector(QQ[x],[1,2,3,4]).pairwise_product(vector(QQ[y],[1,2,3,4])))
        Traceback (most recent call last):
        ...
        TypeError: no common canonical parent for objects with parents: 'Ambient free module of rank 4 over the principal ideal domain Univariate Polynomial Ring in x over Rational Field' and 'Ambient free module of rank 4 over the principal ideal domain Univariate Polynomial Ring in y over Rational Field'
        sage: v = vector({1: 1, 3: 2})  # test sparse vectors
        sage: w = vector({0: 6, 3: -4})
        sage: v.pairwise_product(w)
        (0, 0, 0, -8)
        sage: w.pairwise_product(v) == v.pairwise_product(w)
        True

  `plot`(plot_type=None, start=None, **kwds)

    INPUT:

    - `plot_type` - (default: ‘arrow’ if v has 3 or fewer components,

        otherwise ‘step’) type of plot. Options are:

        - ‘arrow’ to draw an arrow
        - ‘point’ to draw a point at the coordinates specified by the vector
        - ‘step’ to draw a step function representing the coordinates of the vector.

      Both ‘arrow’ and ‘point’ raise exceptions if the vector has more than 3 dimensions.

    - `start` - (default: origin in correct dimension) may be a tuple, list, or vector.

    EXAMPLES:

    The following both plot the given vector:

        sage: v = vector(RDF, (1,2))
        sage: A = plot(v)
        sage: B = v.plot()
        sage: A+B # should just show one vector
        Graphics object consisting of 2 graphics primitives

    Examples of the plot types:

        sage: A = plot(v, plot_type='arrow')
        sage: B = plot(v, plot_type='point', color='green', size=20)
        sage: C = plot(v, plot_type='step') # calls v.plot_step()
        sage: A+B+C
        Graphics object consisting of 3 graphics primitives

    You can use the optional arguments for `plot_step()`:

        sage: eps = 0.1
        sage: plot(v, plot_type='step', eps=eps, xmax=5, hue=0)
        Graphics object consisting of 1 graphics primitive

    Three-dimensional examples:

        sage: v = vector(RDF, (1,2,1))
        sage: plot(v) # defaults to an arrow plot
        Graphics3d Object

        sage: plot(v, plot_type='arrow')
        Graphics3d Object

        sage: from sage.plot.plot3d.shapes2 import frame3d
        sage: plot(v, plot_type='point')+frame3d((0,0,0), v.list())
        Graphics3d Object

        sage: plot(v, plot_type='step') # calls v.plot_step()
        Graphics object consisting of 1 graphics primitive

        sage: plot(v, plot_type='step', eps=eps, xmax=5, hue=0)
        Graphics object consisting of 1 graphics primitive

    With greater than three coordinates, it defaults to a step plot:

        sage: v = vector(RDF, (1,2,3,4))
        sage: plot(v)
        Graphics object consisting of 1 graphics primitive

    One dimensional vectors are plotted along the horizontal axis of the coordinate plane:

        sage: plot(vector([1]))
        Graphics object consisting of 1 graphics primitive

    An optional start argument may also be specified by a tuple, list, or vector:

        sage: u = vector([1,2]); v = vector([2,5])
        sage: plot(u, start=v)
        Graphics object consisting of 1 graphics primitive

    TESTS:

        sage: u = vector([1,1]); v = vector([2,2,2]); z=(3,3,3)
        sage: plot(u) #test when start=None
        Graphics object consisting of 1 graphics primitive

        sage: plot(u, start=v) #test when coordinate dimension mismatch exists
        Traceback (most recent call last):
        ...
        ValueError: vector coordinates are not of the same dimension
        sage: P = plot(v, start=z) #test when start coordinates are passed as a tuple
        sage: P = plot(v, start=list(z)) #test when start coordinates are passed as a list

  `plot_step`(xmin=0, xmax=1, eps=None, res=None, connect=True, **kwds)

    INPUT:

    - `xmin` - (default: 0) start x position to start plotting
    - `xmax` - (default: 1) stop x position to stop plotting
    - `eps` - (default: determined by xmax) we view this vector as defining a function at the points xmin, xmin + eps, xmin + 2*eps, ...,
    - `res` - (default: all points) total number of points to include in the graph
    - `connect` - (default: True) if True draws a line; otherwise draw a list of points.

    EXAMPLES:

        sage: eps=0.1
        sage: v = vector(RDF, [sin(n*eps) for n in range(100)])
        sage: v.plot_step(eps=eps, xmax=5, hue=0)
        Graphics object consisting of 1 graphics primitive

  `row`()

    Return a matrix with a single row and the same entries as the vector `self`.

    OUTPUT:

    A matrix over the same ring as the vector (or free module element), with a single row. The entries of the row are identical to those of the vector, and in the same order.

    EXAMPLES:

        sage: v = vector(ZZ, [1,2,3])
        sage: w = v.row(); w
        [1 2 3]
        sage: w.parent()
        Full MatrixSpace of 1 by 3 dense matrices over Integer Ring

        sage: x = vector(FiniteField(13), [2,4,8,16])
        sage: x.row()
        [2 4 8 3]

    There is more than one way to get one-row matrix from a vector, but the `row` method is more efficient than making a column and then taking a transpose. Notice that supplying a vector to the matrix constructor demonstrates Sage’s preference for rows.

        sage: x = vector(RDF, [sin(i*pi/20) for i in range(10)])
        sage: x.row() == matrix(x)
        True
        sage: x.row() == x.column().transpose()
        True

    Sparse or dense implementations are preserved.

        sage: d = vector(RR, [1.0, 2.0, 3.0])
        sage: s = vector(CDF, {2:5.0+6.0*I})
        sage: dm = d.row()
        sage: sm = s.row()
        sage: all([d.is_dense(), dm.is_dense(), s.is_sparse(), sm.is_sparse()])
        True

    TESTS:

    The `row()` method will return a specified row of a matrix as a vector. So here are a couple of round-trips.

        sage: A = matrix(ZZ, [[1,2,3]])
        sage: A == A.row(0).row()
        True
        sage: v = vector(ZZ, [4,5,6])
        sage: v == v.row().row(0)
        True

    And a very small corner case.

        sage: v = vector(ZZ, [])
        sage: w = v.row()
        sage: w.parent()
        Full MatrixSpace of 1 by 0 dense matrices over Integer Ring

  `set`(i, x)

    The set method is meant to be more efficient than __setitem__, because it need not be guaranteed to do any error checking or coercion. Use with great, great care.

    EXAMPLES:

        sage: v = vector([1/2,2/5,0]); v
        (1/2, 2/5, 0)
        sage: v.set(2, -15/17); v
        (1/2, 2/5, -15/17)

  `set_immutable`()

    Make this vector immutable. This operation can’t be undone.

    EXAMPLES:

        sage: v = vector([1..5]); v
        (1, 2, 3, 4, 5)
        sage: v[1] = 10
        sage: v.set_immutable()
        sage: v[1] = 10
        Traceback (most recent call last):
        ...
        ValueError: vector is immutable; please change a copy instead (use copy())

  `sparse_vector`()

    Return sparse version of self. If self is sparse, just return self; otherwise, create and return correspond sparse vector.

    EXAMPLES:

        sage: vector([-1,0,3,0,0,0]).sparse_vector().is_sparse()
        True
        sage: vector([-1,0,3,0,0,0]).sparse_vector().is_sparse()
        True
        sage: vector([-1,0,3,0,0,0]).sparse_vector()
        (-1, 0, 3, 0, 0, 0)

  `subs`(in_dict=None, **kwds)

    EXAMPLES:

        sage: var('a,b,d,e')
        (a, b, d, e)
        sage: v = vector([a, b, d, e])
        sage: v.substitute(a=1)
        (1, b, d, e)
        sage: v.subs(a=b, b=d)
        (b, d, d, e)

  `support`()

    Return the integers `i` such that `self[i] != 0`. This is the same as the `nonzero_positions` function.

    EXAMPLES:

        sage: vector([-1,0,3,0,0,0,0.01]).support()
        [0, 2, 6]

  `tensor_product`(right)

    Returns a matrix, the outer product of two vectors `self` and `right`.

    INPUT:

    - `right` - a vector (or free module element) of any size, whose elements are compatible (with regard to multiplication) with the elements of `self`.

    OUTPUT:

    The outer product of two vectors \(x\) and \(y\) (respectively `self` and `right`) can be described several ways. If we interpret \(x\) as a \(m\times 1\) matrix and interpret \(y\) as a \(1\times n\) matrix, then the outer product is the \(m\times n\) matrix from the usual matrix product \(xy\). Notice how this is the “opposite” in some ways from an inner product (which would require \(m=n\)).

    If we just consider vectors, use each entry of \(x\) to create a scalar multiples of the vector \(y\) and use these vectors as the rows of a matrix. Or use each entry of \(y\) to create a scalar multiples of \(x\) and use these vectors as the columns of a matrix.

    EXAMPLES:

        sage: u = vector(QQ, [1/2, 1/3, 1/4, 1/5])
        sage: v = vector(ZZ, [60, 180, 600])
        sage: u.outer_product(v)
        [ 30  90 300]
        [ 20  60 200]
        [ 15  45 150]
        [ 12  36 120]
        sage: M = v.outer_product(u); M
        [ 30  20  15  12]
        [ 90  60  45  36]
        [300 200 150 120]
        sage: M.parent()
        Full MatrixSpace of 3 by 4 dense matrices over Rational Field

    The more general `sage.matrix.matrix2.tensor_product()` is an operation on a pair of matrices. If we construe a pair of vectors as a column vector and a row vector, then an outer product and a tensor product are identical. Thus \(tensor_product\) is a synonym for this method.

        sage: u = vector(QQ, [1/2, 1/3, 1/4, 1/5])
        sage: v = vector(ZZ, [60, 180, 600])
        sage: u.tensor_product(v) == (u.column()).tensor_product(v.row())
        True

    The result is always a dense matrix, no matter if the two vectors are, or are not, dense.

        sage: d = vector(ZZ,[4,5], sparse=False)
        sage: s = vector(ZZ, [1,2,3], sparse=True)
        sage: dd = d.outer_product(d)
        sage: ds = d.outer_product(s)
        sage: sd = s.outer_product(d)
        sage: ss = s.outer_product(s)
        sage: all([dd.is_dense(), ds.is_dense(), sd.is_dense(), dd.is_dense()])
        True

    Vectors with no entries do the right thing.

        sage: v = vector(ZZ, [])
        sage: z = v.outer_product(v)
        sage: z.parent()
        Full MatrixSpace of 0 by 0 dense matrices over Integer Ring

    There is a fair amount of latitude in the value of the `right` vector, and the matrix that results can have entries from a new ring large enough to contain the result. If you know better, you can sometimes bring the result down to a less general ring.

        sage: R.<t> = ZZ[]
        sage: v = vector(R, [12, 24*t])
        sage: w = vector(QQ, [1/2, 1/3, 1/4])
        sage: op = v.outer_product(w)
        sage: op
        [   6    4    3]
        [12*t  8*t  6*t]
        sage: op.base_ring()
        Univariate Polynomial Ring in t over Rational Field
        sage: m = op.change_ring(R); m
        [   6    4    3]
        [12*t  8*t  6*t]
        sage: m.base_ring()
        Univariate Polynomial Ring in t over Integer Ring

    But some inputs are not compatible, even if vectors.

        sage: w = vector(GF(5), [1,2])
        sage: v = vector(GF(7), [1,2,3,4])
        sage: z = w.outer_product(v)
        Traceback (most recent call last):
        ...
        TypeError: unsupported operand parent(s) for '*': 'Full MatrixSpace of 2 by 1 dense matrices over Finite Field of size 5' and 'Full MatrixSpace of 1 by 4 dense matrices over Finite Field of size 7'

    And some inputs don’t make any sense at all.

        sage: w=vector(QQ, [5,10])
        sage: z=w.outer_product(6)
        Traceback (most recent call last):
        ...
        TypeError: right operand in an outer product must be a vector, not an element of Integer Ring

  `transpose`()

    Return self as a column matrix.

    Note

    The `transpose()` method has been deprecated as of Sage 4.6.2, in favor of the `column()` method which is functionally identical.

    EXAMPLES:

        sage: v = vector(ZZ, [2, 12, 22])
        sage: transpose(vector(v))
        doctest:...: DeprecationWarning: The transpose() method for vectors has been deprecated, use column() instead
        (or check to see if you have a vector when you really want a matrix)
        See http://trac.sagemath.org/10541 for details.
        [ 2]
        [12]
        [22]

        sage: transpose(vector(GF(7), v))
        [2]
        [5]
        [1]

        sage: transpose(vector(v, ZZ['x', 'y']))
        [ 2]
        [12]
        [22]

class `sage.modules.free_module_element.``FreeModuleElement_generic_dense`

  Bases: `sage.modules.free_module_element.FreeModuleElement`

  A generic dense element of a free module.

  `function`(*args)

    Returns a vector over a callable symbolic expression ring.

    EXAMPLES:

        sage: x,y=var('x,y')
        sage: v=vector([x,y,x*sin(y)])
        sage: w=v.function([x,y]); w
        (x, y) |--> (x, y, x*sin(y))
        sage: w.base_ring()
        Callable function ring with arguments (x, y)
        sage: w(1,2)
        (1, 2, sin(2))
        sage: w(2,1)
        (2, 1, 2*sin(1))
        sage: w(y=1,x=2)
        (2, 1, 2*sin(1))

        sage: x,y=var('x,y')
        sage: v=vector([x,y,x*sin(y)])
        sage: w=v.function([x]); w
        x |--> (x, y, x*sin(y))
        sage: w.base_ring()
        Callable function ring with arguments (x,)
        sage: w(4)
        (4, y, 4*sin(y))

  `list`(copy=True)

    Return list of elements of self.

    INPUT:

    > - copy – bool, return list of underlying entries

    EXAMPLES:

        sage: P.<x,y,z> = QQ[]
        sage: v = vector([x,y,z])
        sage: type(v)
        <type 'sage.modules.free_module_element.FreeModuleElement_generic_dense'>
        sage: a = v.list(); a
        [x, y, z]
        sage: a[0] = x*y; v
        (x, y, z)
        sage: a = v.list(copy=False); a
        [x, y, z]
        sage: a[0] = x*y; v
        (x*y, y, z)

class `sage.modules.free_module_element.``FreeModuleElement_generic_sparse`

  Bases: `sage.modules.free_module_element.FreeModuleElement`

  A generic sparse free module element is a dictionary with keys ints i and entries in the base ring.

  EXAMPLES:

  Pickling works:

      sage: v = FreeModule(ZZ, 3, sparse=True).0
      sage: loads(dumps(v)) == v
      True
      sage: v = FreeModule(Integers(8)['x,y'], 5, sparse=True).1
      sage: loads(dumps(v)) - v
      (0, 0, 0, 0, 0)

      sage: a = vector([-1,0,1/1],sparse=True); b = vector([-1/1,0,0],sparse=True)
      sage: a.parent()
      Sparse vector space of dimension 3 over Rational Field
      sage: b - a
      (0, 0, -1)
      sage: (b-a).dict()
      {2: -1}

  `denominator`()

    Return the least common multiple of the denominators of the entries of self.

    EXAMPLES:

        sage: v = vector([1/2,2/5,3/14], sparse=True)
        sage: v.denominator()
        70

  `dict`(copy=True)

    Return dictionary of nonzero entries of self.

    INPUT:

    > - `copy` – bool (default: True)

    OUTPUT:

    > - Python dictionary

    EXAMPLES:

        sage: v = vector([0,0,0,0,1/2,0,3/14], sparse=True)
        sage: v.dict()
        {4: 1/2, 6: 3/14}

  `get`(i)

    Like __getitem__ but with no guaranteed type or bounds checking. Returns 0 if access is out of bounds.

    EXAMPLES:

        sage: v = vector([1,2/3,pi], sparse=True)
        sage: v.get(1)
        2/3
        sage: v.get(10)
        0

  `hamming_weight`()

    Returns the number of positions `i` such that `self[i] != 0`.

    EXAMPLES:

        sage: v = vector({1: 1, 3: -2})
        sage: w = vector({1: 4, 3: 2})
        sage: v+w
        (0, 5, 0, 0)
        sage: (v+w).hamming_weight()
        1

  `iteritems`()

    Return iterator over the entries of self.

    EXAMPLES:

        sage: v = vector([1,2/3,pi], sparse=True)
        sage: v.iteritems()
        <dictionary-itemiterator object at ...>
        sage: list(v.iteritems())
        [(0, 1), (1, 2/3), (2, pi)]

  `list`(copy=True)

    Return list of elements of self.

    INPUT:

    > - copy – bool, return list of underlying entries

    EXAMPLES:

        sage: v = vector([1,2/3,pi], sparse=True)
        sage: type(v)
        <type 'sage.modules.free_module_element.FreeModuleElement_generic_sparse'>
        sage: a = v.list(); a
        [1, 2/3, pi]

  `nonzero_positions`()

    Returns the list of numbers `i` such that `self[i] != 0`.

    EXAMPLES:

        sage: v = vector({1: 1, 3: -2})
        sage: w = vector({1: 4, 3: 2})
        sage: v+w
        (0, 5, 0, 0)
        sage: (v+w).nonzero_positions()
        [1]

  `set`(i, x)

    Like __setitem__ but with no guaranteed type or bounds checking.

    EXAMPLES:

        sage: v = vector([1,2/3,pi], sparse=True)
        sage: v.set(1, pi^3)
        sage: v
        (1, pi^3, pi)

    No bounds checking:

        sage: v.set(10, pi)

    This lack of bounds checking causes trouble later:

        sage: v
        <repr(<sage.modules.free_module_element.FreeModuleElement_generic_sparse at 0x...>) failed: IndexError: list assignment index out of range>

`sage.modules.free_module_element.``free_module_element`(arg0, arg1=None, arg2=None, sparse=None)

  Return a vector or free module element with specified entries.

  CALL FORMATS:

  This constructor can be called in several different ways. In each case, `sparse=True` or `sparse=False` can be supplied as an option. `free_module_element()` is an alias for `vector()`.

  > 1. vector(object)
  > 2. vector(ring, object)
  > 3. vector(object, ring)
  > 4. vector(ring, degree, object)
  > 5. vector(ring, degree)
  > 6. vector(numpy_array)

  INPUT:

  - `object` - a list, dictionary, or other iterable containing the entries of the vector, including any object that is palatable to the `Sequence` constructor
  - `ring` - a base ring (or field) for the vector space or free module, which contains all of the elements
  - `degree` - an integer specifying the number of entries in the vector or free module element
  - `numpy_array` - a NumPy array with the desired entries
  - `sparse` - optional

  In call format 4, an error is raised if the `degree` does not match the length of `object` so this call can provide some safeguards. Note however that using this format when `object` is a dictionary is unlikely to work properly.

  OUTPUT:

  An element of the vector space or free module with the given base ring and implied or specified dimension or rank, containing the specified entries and with correct degree.

  In call format 5, no entries are specified, so the element is populated with all zeros.

  If the `sparse` option is not supplied, the output will generally have a dense representation. The exception is if `object` is a dictionary, then the representation will be sparse.

  EXAMPLES:

      sage: v = vector([1,2,3]); v
      (1, 2, 3)
      sage: v.parent()
      Ambient free module of rank 3 over the principal ideal domain Integer Ring
      sage: v = vector([1,2,3/5]); v
      (1, 2, 3/5)
      sage: v.parent()
      Vector space of dimension 3 over Rational Field

  All entries must canonically coerce to some common ring:

      sage: v = vector([17, GF(11)(5), 19/3]); v
      Traceback (most recent call last):
      ...
      TypeError: unable to find a common ring for all elements

      sage: v = vector([17, GF(11)(5), 19]); v
      (6, 5, 8)
      sage: v.parent()
      Vector space of dimension 3 over Finite Field of size 11
      sage: v = vector([17, GF(11)(5), 19], QQ); v
      (17, 5, 19)
      sage: v.parent()
      Vector space of dimension 3 over Rational Field
      sage: v = vector((1,2,3), QQ); v
      (1, 2, 3)
      sage: v.parent()
      Vector space of dimension 3 over Rational Field
      sage: v = vector(QQ, (1,2,3)); v
      (1, 2, 3)
      sage: v.parent()
      Vector space of dimension 3 over Rational Field
      sage: v = vector(vector([1,2,3])); v
      (1, 2, 3)
      sage: v.parent()
      Ambient free module of rank 3 over the principal ideal domain Integer Ring

  You can also use `free_module_element`, which is the same as `vector`.

      sage: free_module_element([1/3, -4/5])
      (1/3, -4/5)

  We make a vector mod 3 out of a vector over \(\ZZ\).

      sage: vector(vector([1,2,3]), GF(3))
      (1, 2, 0)

  The degree of a vector may be specified:

      sage: vector(QQ, 4, [1,1/2,1/3,1/4])
      (1, 1/2, 1/3, 1/4)

  But it is an error if the degree and size of the list of entries are mismatched:

      sage: vector(QQ, 5, [1,1/2,1/3,1/4])
      Traceback (most recent call last):
      ...
      ValueError: incompatible degrees in vector constructor

  Providing no entries populates the vector with zeros, but of course, you must specify the degree since it is not implied. Here we use a finite field as the base ring.

      sage: w = vector(FiniteField(7), 4); w
      (0, 0, 0, 0)
      sage: w.parent()
      Vector space of dimension 4 over Finite Field of size 7

  The fastest method to construct a zero vector is to call the `zero_vector()` method directly on a free module or vector space, since vector(...) must do a small amount of type checking. Almost as fast as the `zero_vector()` method is the `zero_vector()` constructor, which defaults to the integers.

      sage: vector(ZZ, 5)          # works fine
      (0, 0, 0, 0, 0)
      sage: (ZZ^5).zero_vector()   # very tiny bit faster
      (0, 0, 0, 0, 0)
      sage: zero_vector(ZZ, 5)     # similar speed to vector(...)
      (0, 0, 0, 0, 0)
      sage: z = zero_vector(5); z
      (0, 0, 0, 0, 0)
      sage: z.parent()
      Ambient free module of rank 5 over
      the principal ideal domain Integer Ring

  Here we illustrate the creation of sparse vectors by using a dictionary.

      sage: vector({1:1.1, 3:3.14})
      (0.000000000000000, 1.10000000000000, 0.000000000000000, 3.14000000000000)

  With no degree given, a dictionary of entries implicitly declares a degree by the largest index (key) present. So you can provide a terminal element (perhaps a zero?) to set the degree. But it is probably safer to just include a degree in your construction.

      sage: v = vector(QQ, {0:1/2, 4:-6, 7:0}); v
      (1/2, 0, 0, 0, -6, 0, 0, 0)
      sage: v.degree()
      8
      sage: v.is_sparse()
      True
      sage: w = vector(QQ, 8, {0:1/2, 4:-6})
      sage: w == v
      True

  It is an error to specify a negative degree.

      sage: vector(RR, -4, [1.0, 2.0, 3.0, 4.0])
      Traceback (most recent call last):
      ...
      ValueError: cannot specify the degree of a vector as a negative integer (-4)

  It is an error to create a zero vector but not provide a ring as the first argument.

      sage: vector('junk', 20)
      Traceback (most recent call last):
      ...
      TypeError: first argument must be base ring of zero vector, not junk

  And it is an error to specify an index in a dictionary that is greater than or equal to a requested degree.

      sage: vector(ZZ, 10, {3:4, 7:-2, 10:637})
      Traceback (most recent call last):
      ...
      ValueError: dictionary of entries has a key (index) exceeding the requested degree

  Any 1 dimensional numpy array of type float or complex may be passed to vector. The result will be a vector in the appropriate dimensional vector space over the real double field or the complex double field. The data in the array must be contiguous so column-wise slices of numpy matrices will raise an exception.

      sage: import numpy
      sage: x=numpy.random.randn(10)
      sage: y=vector(x)
      sage: v=numpy.random.randn(10)*numpy.complex(0,1)
      sage: w=vector(v)

  If any of the arguments to vector have Python type int, long, real, or complex, they will first be coerced to the appropriate Sage objects. This fixes trac #3847.

      sage: v = vector([int(0)]); v
      (0)
      sage: v[0].parent()
      Integer Ring
      sage: v = vector(range(10)); v
      (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)
      sage: v[3].parent()
      Integer Ring
      sage: v = vector([float(23.4), int(2), complex(2+7*I), long(1)]); v
      (23.4, 2.0, 2.0 + 7.0*I, 1.0)
      sage: v[1].parent()
      Complex Double Field

  If the argument is a vector, it doesn’t change the base ring. This fixes trac #6643.

      sage: K.<sqrt3> = QuadraticField(3)
      sage: u = vector(K, (1/2, sqrt3/2) )
      sage: vector(u).base_ring()
      Number Field in sqrt3 with defining polynomial x^2 - 3
      sage: v = vector(K, (0, 1) )
      sage: vector(v).base_ring()
      Number Field in sqrt3 with defining polynomial x^2 - 3

  Constructing a vector from a numpy array behaves as expected:

      sage: import numpy
      sage: a=numpy.array([1,2,3])
      sage: v=vector(a); v
      (1, 2, 3)
      sage: parent(v)
      Ambient free module of rank 3 over the principal ideal domain Integer Ring

  Complex numbers can be converted naturally to a sequence of length 2. And then to a vector.

      sage: c = CDF(2 + 3*I)
      sage: v = vector(c); v
      (2.0, 3.0)

  A generator, or other iterable, may also be supplied as input. Anything that can be converted to a `Sequence` is a possible input.

      sage: type(i^2 for i in range(3))
      <type 'generator'>
      sage: v = vector(i^2 for i in range(3)); v
      (0, 1, 4)

  An empty list, without a ring given, will default to the integers.

      sage: x = vector([]); x
      ()
      sage: x.parent()
      Ambient free module of rank 0 over the principal ideal domain Integer Ring

`sage.modules.free_module_element.``is_FreeModuleElement`(x)

  EXAMPLES:

      sage: sage.modules.free_module_element.is_FreeModuleElement(0)
      False
      sage: sage.modules.free_module_element.is_FreeModuleElement(vector([1,2,3]))
      True

`sage.modules.free_module_element.``make_FreeModuleElement_generic_dense`(parent, entries, degree)

  EXAMPLES:

      sage: sage.modules.free_module_element.make_FreeModuleElement_generic_dense(QQ^3, [1,2,-3/7], 3)
      (1, 2, -3/7)

`sage.modules.free_module_element.``make_FreeModuleElement_generic_dense_v1`(parent, entries, degree, is_mutable)

  EXAMPLES:

      sage: v = sage.modules.free_module_element.make_FreeModuleElement_generic_dense_v1(QQ^3, [1,2,-3/7], 3, True); v
      (1, 2, -3/7)
      sage: v[0] = 10; v
      (10, 2, -3/7)
      sage: v = sage.modules.free_module_element.make_FreeModuleElement_generic_dense_v1(QQ^3, [1,2,-3/7], 3, False); v
      (1, 2, -3/7)
      sage: v[0] = 10
      Traceback (most recent call last):
      ...
      ValueError: vector is immutable; please change a copy instead (use copy())

`sage.modules.free_module_element.``make_FreeModuleElement_generic_sparse`(parent, entries, degree)

  EXAMPLES:

      sage: v = sage.modules.free_module_element.make_FreeModuleElement_generic_sparse(QQ^3, {2:5/2}, 3); v
      (0, 0, 5/2)

`sage.modules.free_module_element.``make_FreeModuleElement_generic_sparse_v1`(parent, entries, degree, is_mutable)

  EXAMPLES:

      sage: v = sage.modules.free_module_element.make_FreeModuleElement_generic_sparse_v1(QQ^3, {2:5/2}, 3, False); v
      (0, 0, 5/2)
      sage: v.is_mutable()
      False

`sage.modules.free_module_element.``prepare`(v, R, degree=None)

  Converts an object describing elements of a vector into a list of entries in a common ring.

  INPUT:

  - `v` - a dictionary with non-negative integers as keys, or a list or other object that can be converted by the `Sequence` constructor
  - `R` - a ring containing all the entries, possibly given as `None`
  - `degree` - a requested size for the list when the input is a dictionary, otherwise ignored

  OUTPUT:

  A pair.

  The first item is a list of the values specified in the object `v`. If the object is a dictionary , entries are placed in the list according to the indices that were their keys in the dictionary, and the remainder of the entries are zero. The value of `degree` is assumed to be larger than any index provided in the dictionary and will be used as the number of entries in the returned list.

  The second item returned is a ring that contains all of the entries in the list. If `R` is given, the entries are coerced in. Otherwise a common ring is found. For more details, see the `Sequence` object. When `v` has no elements and `R` is `None`, the ring returned is the integers.

  EXAMPLES:

      sage: from sage.modules.free_module_element import prepare
      sage: prepare([1,2/3,5],None)
      ([1, 2/3, 5], Rational Field)

      sage: prepare([1,2/3,5],RR)
      ([1.00000000000000, 0.666666666666667, 5.00000000000000], Real Field with 53 bits of precision)

      sage: prepare({1:4, 3:-2}, ZZ, 6)
      ([0, 4, 0, -2, 0, 0], Integer Ring)

      sage: prepare({3:1, 5:3}, QQ, 6)
      ([0, 0, 0, 1, 0, 3], Rational Field)

      sage: prepare([1,2/3,'10',5],RR)
      ([1.00000000000000, 0.666666666666667, 10.0000000000000, 5.00000000000000], Real Field with 53 bits of precision)

      sage: prepare({},QQ, 0)
      ([], Rational Field)

      sage: prepare([1,2/3,'10',5],None)
      Traceback (most recent call last):
      ...
      TypeError: unable to find a common ring for all elements

  Some objects can be converted to sequences even if they are not always thought of as sequences.

      sage: c = CDF(2+3*I)
      sage: prepare(c, None)
      ([2.0, 3.0], Real Double Field)

  This checks a bug listed at Trac #10595. Without good evidence for a ring, the default is the integers.

      sage: prepare([], None)
      ([], Integer Ring)

`sage.modules.free_module_element.``random_vector`(ring, degree=None, *args, **kwds)

  Returns a vector (or module element) with random entries.

  INPUT:

  - ring - default: `ZZ` - the base ring for the entries
  - degree - a non-negative integer for the number of entries in the vector
  - sparse - default: `False` - whether to use a sparse implementation
  - args, kwds - additional arguments and keywords are passed to the `random_element()` method of the ring

  OUTPUT:

  A vector, or free module element, with `degree` elements from `ring`, chosen randomly from the ring according to the ring’s `random_element()` method.

  Note

  See below for examples of how random elements are generated by some common base rings.

  EXAMPLES:

  First, module elements over the integers. The default distribution is tightly clustered around -1, 0, 1. Uniform distributions can be specified by giving bounds, though the upper bound is never met. See `sage.rings.integer_ring.IntegerRing_class.random_element()` for several other variants.

      sage: random_vector(10)
      (-8, 2, 0, 0, 1, -1, 2, 1, -95, -1)

      sage: sorted(random_vector(20))
      [-12, -6, -4, -4, -2, -2, -2, -1, -1, -1, 0, 0, 0, 0, 0, 1, 1, 1, 4, 5]

      sage: random_vector(ZZ, 20, x=4)
      (2, 0, 3, 0, 1, 0, 2, 0, 2, 3, 0, 3, 1, 2, 2, 2, 1, 3, 2, 3)

      sage: random_vector(ZZ, 20, x=-20, y=100)
      (43, 47, 89, 31, 56, -20, 23, 52, 13, 53, 49, -12, -2, 94, -1, 95, 60, 83, 28, 63)

      sage: random_vector(ZZ, 20, distribution="1/n")
      (0, -1, -2, 0, -1, -2, 0, 0, 27, -1, 1, 1, 0, 2, -1, 1, -1, -2, -1, 3)

  If the ring is not specified, the default is the integers, and parameters for the random distribution may be passed without using keywords. This is a random vector with 20 entries uniformly distributed between -20 and 100.

      sage: random_vector(20, -20, 100)
      (70, 19, 98, 2, -18, 88, 36, 66, 76, 52, 82, 99, 55, -17, 82, -15, 36, 28, 79, 18)

  Now over the rationals. Note that bounds on the numerator and denominator may be specified. See `sage.rings.rational_field.RationalField.random_element()` for documentation.

      sage: random_vector(QQ, 10)
      (0, -1, -4/3, 2, 0, -13, 2/3, 0, -4/5, -1)

      sage: random_vector(QQ, 10, num_bound = 15, den_bound = 5)
      (-12/5, 9/4, -13/3, -1/3, 1, 5/4, 4, 1, -15, 10/3)

  Inexact rings may be used as well. The reals have uniform distributions, with the range \((-1,1)\) as the default. More at: `sage.rings.real_mpfr.RealField_class.random_element()`

      sage: random_vector(RR, 5)
      (0.248997268533725, -0.112200126330480, 0.776829203293064, -0.899146461031406, 0.534465018743125)

      sage: random_vector(RR, 5, min = 8, max = 14)
      (8.43260944052606, 8.34129413391087, 8.92391495103829, 11.5784799413416, 11.0973561568002)

  Any ring with a `random_element()` method may be used.

      sage: F = FiniteField(23)
      sage: hasattr(F, 'random_element')
      True
      sage: random_vector(F, 10)
      (21, 6, 5, 2, 6, 2, 18, 9, 9, 7)

  The default implementation is a dense representation, equivalent to setting `sparse=False`.

      sage: v = random_vector(10)
      sage: v.is_sparse()
      False

      sage: w = random_vector(ZZ, 20, sparse=True)
      sage: w.is_sparse()
      True

  Inputs get checked before constructing the vector.

      sage: random_vector('junk')
      Traceback (most recent call last):
      ...
      TypeError: degree of a random vector must be an integer, not None

      sage: random_vector('stuff', 5)
      Traceback (most recent call last):
      ...
      TypeError: elements of a vector, or module element, must come from a ring, not stuff

      sage: random_vector(ZZ, -9)
      Traceback (most recent call last):
      ...
      ValueError: degree of a random vector must be non-negative, not -9

`sage.modules.free_module_element.``vector`(arg0, arg1=None, arg2=None, sparse=None)

  Return a vector or free module element with specified entries.

  CALL FORMATS:

  This constructor can be called in several different ways. In each case, `sparse=True` or `sparse=False` can be supplied as an option. `free_module_element()` is an alias for `vector()`.

  > 1. vector(object)
  > 2. vector(ring, object)
  > 3. vector(object, ring)
  > 4. vector(ring, degree, object)
  > 5. vector(ring, degree)
  > 6. vector(numpy_array)

  INPUT:

  - `object` - a list, dictionary, or other iterable containing the entries of the vector, including any object that is palatable to the `Sequence` constructor
  - `ring` - a base ring (or field) for the vector space or free module, which contains all of the elements
  - `degree` - an integer specifying the number of entries in the vector or free module element
  - `numpy_array` - a NumPy array with the desired entries
  - `sparse` - optional

  In call format 4, an error is raised if the `degree` does not match the length of `object` so this call can provide some safeguards. Note however that using this format when `object` is a dictionary is unlikely to work properly.

  OUTPUT:

  An element of the vector space or free module with the given base ring and implied or specified dimension or rank, containing the specified entries and with correct degree.

  In call format 5, no entries are specified, so the element is populated with all zeros.

  If the `sparse` option is not supplied, the output will generally have a dense representation. The exception is if `object` is a dictionary, then the representation will be sparse.

  EXAMPLES:

      sage: v = vector([1,2,3]); v
      (1, 2, 3)
      sage: v.parent()
      Ambient free module of rank 3 over the principal ideal domain Integer Ring
      sage: v = vector([1,2,3/5]); v
      (1, 2, 3/5)
      sage: v.parent()
      Vector space of dimension 3 over Rational Field

  All entries must canonically coerce to some common ring:

      sage: v = vector([17, GF(11)(5), 19/3]); v
      Traceback (most recent call last):
      ...
      TypeError: unable to find a common ring for all elements

      sage: v = vector([17, GF(11)(5), 19]); v
      (6, 5, 8)
      sage: v.parent()
      Vector space of dimension 3 over Finite Field of size 11
      sage: v = vector([17, GF(11)(5), 19], QQ); v
      (17, 5, 19)
      sage: v.parent()
      Vector space of dimension 3 over Rational Field
      sage: v = vector((1,2,3), QQ); v
      (1, 2, 3)
      sage: v.parent()
      Vector space of dimension 3 over Rational Field
      sage: v = vector(QQ, (1,2,3)); v
      (1, 2, 3)
      sage: v.parent()
      Vector space of dimension 3 over Rational Field
      sage: v = vector(vector([1,2,3])); v
      (1, 2, 3)
      sage: v.parent()
      Ambient free module of rank 3 over the principal ideal domain Integer Ring

  You can also use `free_module_element`, which is the same as `vector`.

      sage: free_module_element([1/3, -4/5])
      (1/3, -4/5)

  We make a vector mod 3 out of a vector over \(\ZZ\).

      sage: vector(vector([1,2,3]), GF(3))
      (1, 2, 0)

  The degree of a vector may be specified:

      sage: vector(QQ, 4, [1,1/2,1/3,1/4])
      (1, 1/2, 1/3, 1/4)

  But it is an error if the degree and size of the list of entries are mismatched:

      sage: vector(QQ, 5, [1,1/2,1/3,1/4])
      Traceback (most recent call last):
      ...
      ValueError: incompatible degrees in vector constructor

  Providing no entries populates the vector with zeros, but of course, you must specify the degree since it is not implied. Here we use a finite field as the base ring.

      sage: w = vector(FiniteField(7), 4); w
      (0, 0, 0, 0)
      sage: w.parent()
      Vector space of dimension 4 over Finite Field of size 7

  The fastest method to construct a zero vector is to call the `zero_vector()` method directly on a free module or vector space, since vector(...) must do a small amount of type checking. Almost as fast as the `zero_vector()` method is the `zero_vector()` constructor, which defaults to the integers.

      sage: vector(ZZ, 5)          # works fine
      (0, 0, 0, 0, 0)
      sage: (ZZ^5).zero_vector()   # very tiny bit faster
      (0, 0, 0, 0, 0)
      sage: zero_vector(ZZ, 5)     # similar speed to vector(...)
      (0, 0, 0, 0, 0)
      sage: z = zero_vector(5); z
      (0, 0, 0, 0, 0)
      sage: z.parent()
      Ambient free module of rank 5 over
      the principal ideal domain Integer Ring

  Here we illustrate the creation of sparse vectors by using a dictionary.

      sage: vector({1:1.1, 3:3.14})
      (0.000000000000000, 1.10000000000000, 0.000000000000000, 3.14000000000000)

  With no degree given, a dictionary of entries implicitly declares a degree by the largest index (key) present. So you can provide a terminal element (perhaps a zero?) to set the degree. But it is probably safer to just include a degree in your construction.

      sage: v = vector(QQ, {0:1/2, 4:-6, 7:0}); v
      (1/2, 0, 0, 0, -6, 0, 0, 0)
      sage: v.degree()
      8
      sage: v.is_sparse()
      True
      sage: w = vector(QQ, 8, {0:1/2, 4:-6})
      sage: w == v
      True

  It is an error to specify a negative degree.

      sage: vector(RR, -4, [1.0, 2.0, 3.0, 4.0])
      Traceback (most recent call last):
      ...
      ValueError: cannot specify the degree of a vector as a negative integer (-4)

  It is an error to create a zero vector but not provide a ring as the first argument.

      sage: vector('junk', 20)
      Traceback (most recent call last):
      ...
      TypeError: first argument must be base ring of zero vector, not junk

  And it is an error to specify an index in a dictionary that is greater than or equal to a requested degree.

      sage: vector(ZZ, 10, {3:4, 7:-2, 10:637})
      Traceback (most recent call last):
      ...
      ValueError: dictionary of entries has a key (index) exceeding the requested degree

  Any 1 dimensional numpy array of type float or complex may be passed to vector. The result will be a vector in the appropriate dimensional vector space over the real double field or the complex double field. The data in the array must be contiguous so column-wise slices of numpy matrices will raise an exception.

      sage: import numpy
      sage: x=numpy.random.randn(10)
      sage: y=vector(x)
      sage: v=numpy.random.randn(10)*numpy.complex(0,1)
      sage: w=vector(v)

  If any of the arguments to vector have Python type int, long, real, or complex, they will first be coerced to the appropriate Sage objects. This fixes trac #3847.

      sage: v = vector([int(0)]); v
      (0)
      sage: v[0].parent()
      Integer Ring
      sage: v = vector(range(10)); v
      (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)
      sage: v[3].parent()
      Integer Ring
      sage: v = vector([float(23.4), int(2), complex(2+7*I), long(1)]); v
      (23.4, 2.0, 2.0 + 7.0*I, 1.0)
      sage: v[1].parent()
      Complex Double Field

  If the argument is a vector, it doesn’t change the base ring. This fixes trac #6643.

      sage: K.<sqrt3> = QuadraticField(3)
      sage: u = vector(K, (1/2, sqrt3/2) )
      sage: vector(u).base_ring()
      Number Field in sqrt3 with defining polynomial x^2 - 3
      sage: v = vector(K, (0, 1) )
      sage: vector(v).base_ring()
      Number Field in sqrt3 with defining polynomial x^2 - 3

  Constructing a vector from a numpy array behaves as expected:

      sage: import numpy
      sage: a=numpy.array([1,2,3])
      sage: v=vector(a); v
      (1, 2, 3)
      sage: parent(v)
      Ambient free module of rank 3 over the principal ideal domain Integer Ring

  Complex numbers can be converted naturally to a sequence of length 2. And then to a vector.

      sage: c = CDF(2 + 3*I)
      sage: v = vector(c); v
      (2.0, 3.0)

  A generator, or other iterable, may also be supplied as input. Anything that can be converted to a `Sequence` is a possible input.

      sage: type(i^2 for i in range(3))
      <type 'generator'>
      sage: v = vector(i^2 for i in range(3)); v
      (0, 1, 4)

  An empty list, without a ring given, will default to the integers.

      sage: x = vector([]); x
      ()
      sage: x.parent()
      Ambient free module of rank 0 over the principal ideal domain Integer Ring

`sage.modules.free_module_element.``zero_vector`(arg0, arg1=None)

  Returns a vector or free module element with a specified number of zeros.

  CALL FORMATS:

  > 1. zero_vector(degree)
  > 2. zero_vector(ring, degree)

  INPUT:

  - `degree` - the number of zero entries in the vector or free module element
  - `ring` - default `ZZ` - the base ring of the vector space or module containing the constructed zero vector

  OUTPUT:

  A vector or free module element with `degree` entries, all equal to zero and belonging to the ring if specified. If no ring is given, a free module element over `ZZ` is returned.

  EXAMPLES:

  A zero vector over the field of rationals.

      sage: v = zero_vector(QQ, 5); v
      (0, 0, 0, 0, 0)
      sage: v.parent()
      Vector space of dimension 5 over Rational Field

  A free module zero element.

      sage: w = zero_vector(Integers(6), 3); w
      (0, 0, 0)
      sage: w.parent()
      Ambient free module of rank 3 over Ring of integers modulo 6

  If no ring is given, the integers are used.

      sage: u = zero_vector(9); u
      (0, 0, 0, 0, 0, 0, 0, 0, 0)
      sage: u.parent()
      Ambient free module of rank 9 over the principal ideal domain Integer Ring

  Non-integer degrees produce an error.

      sage: zero_vector(5.6)
      Traceback (most recent call last):
      ...
      TypeError: Attempt to coerce non-integral RealNumber to Integer

  Negative degrees also give an error.

      sage: zero_vector(-3)
      Traceback (most recent call last):
      ...
      ValueError: rank (=-3) must be nonnegative

  Garbage instead of a ring will be recognized as such.

      sage: zero_vector(x^2, 5)
      Traceback (most recent call last):
      ...
      TypeError: first argument must be a ring

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