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# Vectors over callable symbolic rings

AUTHOR:
  – Jason Grout (2010)

EXAMPLES:

    sage: f(r, theta, z) = (r*cos(theta), r*sin(theta), z)
    sage: f.parent()
    Vector space of dimension 3 over Callable function ring with arguments (r, theta, z)
    sage: f
    (r, theta, z) |--> (r*cos(theta), r*sin(theta), z)
    sage: f[0]
    (r, theta, z) |--> r*cos(theta)
    sage: f+f
    (r, theta, z) |--> (2*r*cos(theta), 2*r*sin(theta), 2*z)
    sage: 3*f
    (r, theta, z) |--> (3*r*cos(theta), 3*r*sin(theta), 3*z)
    sage: f*f # dot product
    (r, theta, z) |--> r^2*cos(theta)^2 + r^2*sin(theta)^2 + z^2
    sage: f.diff()(0,1,2) # the matrix derivative
    [cos(1)      0      0]
    [sin(1)      0      0]
    [     0      0      1]

TESTS:

    sage: f(u,v,w) = (2*u+v,u-w,w^2+u)
    sage: loads(dumps(f)) == f
    True

class `sage.modules.vector_callable_symbolic_dense.``Vector_callable_symbolic_dense`

  Bases: `sage.modules.free_module_element.FreeModuleElement_generic_dense`

  EXAMPLES:

      sage: type(vector([-1,0,3,pi]))   # indirect doctest
      <class 'sage.modules.vector_symbolic_dense.Vector_symbolic_dense'>

  TESTS:

  Check that #11751 is fixed:

      sage: K.<x> = QQ[]
      sage: M = K^1
      sage: N = M.span([[1/x]]); N
      Free module of degree 1 and rank 1 over Univariate Polynomial Ring in x over Rational Field
      Echelon basis matrix:
      [1/x]
      sage: N([1/x]) # this used to fail prior to #11751
      (1/x)
      sage: N([1/x^2])
      Traceback (most recent call last):
      ...
      TypeError: element (= [1/x^2]) is not in free module

      sage: L=K^2
      sage: R=L.span([[x,0],[0,1/x]], check=False, already_echelonized=True)
      sage: R.basis()[0][0].parent()
      Fraction Field of Univariate Polynomial Ring in x over Rational Field
      sage: R=L.span([[x,x^2]])
      sage: R.basis()[0][0].parent()
      Univariate Polynomial Ring in x over Rational Field

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