예제 #1
0
def linear_transformation(arg0, arg1=None, arg2=None, side='left'):
    r"""
    Create a linear transformation from a variety of possible inputs.

    FORMATS:

    In the following, ``D`` and ``C`` are vector spaces over
    the same field that are the domain and codomain
    (respectively) of the linear transformation.

    ``side`` is a keyword that is either 'left' or 'right'.
    When a matrix is used to specify a linear transformation,
    as in the first two call formats below, you may specify
    if the function is given by matrix multiplication with
    the vector on the left, or the vector on the right.
    The default is 'left'. Internally representations are
    always carried as the 'left' version, and the default
    text representation is this version.  However, the matrix
    representation may be obtained as either version, no matter
    how it is created.

    - ``linear_transformation(A, side='left')``

      Where ``A`` is a matrix.  The domain and codomain are inferred
      from the dimension of the matrix and the base ring of the matrix.
      The base ring must be a field, or have its fraction field implemented
      in Sage.

    - ``linear_transformation(D, C, A, side='left')``

      ``A`` is a matrix that behaves as above.  However, now the domain
      and codomain are given explicitly. The matrix is checked for
      compatibility with the domain and codomain.  Additionally, the
      domain and codomain may be supplied with alternate ("user") bases
      and the matrix is interpreted as being a representation relative
      to those bases.

    - ``linear_transformation(D, C, f)``

      ``f`` is any function that can be applied to the basis elements of the
      domain and that produces elements of the codomain.  The linear
      transformation returned is the unique linear transformation that
      extends this mapping on the basis elements.  ``f`` may come from a
      function defined by a Python ``def`` statement, or may be defined as a
      ``lambda`` function.

      Alternatively, ``f`` may be specified by a callable symbolic function,
      see the examples below for a demonstration.

    - ``linear_transformation(D, C, images)``

      ``images`` is a list, or tuple, of codomain elements, equal in number
      to the size of the basis of the domain.  Each basis element of the domain
      is mapped to the corresponding element of the ``images`` list, and the
      linear transformation returned is the unique linear transfromation that
      extends this mapping.

    OUTPUT:

    A linear transformation described by the input.  This is a
    "vector space morphism", an object of the class
    :class:`sage.modules.vector_space_morphism`.

    EXAMPLES:

    We can define a linear transformation with just a matrix, understood to
    act on a vector placed on one side or the other.  The field for the 
    vector spaces used as domain and codomain is obtained from the base 
    ring of the matrix, possibly promoting to a fraction field.  ::

        sage: A = matrix(ZZ, [[1, -1, 4], [2, 0, 5]])
        sage: phi = linear_transformation(A)
        sage: phi
        Vector space morphism represented by the matrix:
        [ 1 -1  4]
        [ 2  0  5]
        Domain: Vector space of dimension 2 over Rational Field
        Codomain: Vector space of dimension 3 over Rational Field
        sage: phi([1/2, 5])
        (21/2, -1/2, 27)

        sage: B = matrix(Integers(7), [[1, 2, 1], [3, 5, 6]])
        sage: rho = linear_transformation(B, side='right')
        sage: rho
        Vector space morphism represented by the matrix:
        [1 3]
        [2 5]
        [1 6]
        Domain: Vector space of dimension 3 over Ring of integers modulo 7
        Codomain: Vector space of dimension 2 over Ring of integers modulo 7
        sage: rho([2, 4, 6])
        (2, 6)

    We can define a linear transformation with a matrix, while explicitly
    giving the domain and codomain.  Matrix entries will be coerced into the
    common field of scalars for the vector spaces.  ::

        sage: D = QQ^3
        sage: C = QQ^2
        sage: A = matrix([[1, 7], [2, -1], [0, 5]])
        sage: A.parent()
        Full MatrixSpace of 3 by 2 dense matrices over Integer Ring
        sage: zeta = linear_transformation(D, C, A)
        sage: zeta.matrix().parent()
        Full MatrixSpace of 3 by 2 dense matrices over Rational Field
        sage: zeta
        Vector space morphism represented by the matrix:
        [ 1  7]
        [ 2 -1]
        [ 0  5]
        Domain: Vector space of dimension 3 over Rational Field
        Codomain: Vector space of dimension 2 over Rational Field

    Matrix representations are relative to the bases for the domain
    and codomain.  ::

        sage: u = vector(QQ, [1, -1])
        sage: v = vector(QQ, [2, 3])
        sage: D = (QQ^2).subspace_with_basis([u, v])
        sage: x = vector(QQ, [2, 1])
        sage: y = vector(QQ, [-1, 4])
        sage: C = (QQ^2).subspace_with_basis([x, y])
        sage: A = matrix(QQ, [[2, 5], [3, 7]])
        sage: psi = linear_transformation(D, C, A)
        sage: psi
        Vector space morphism represented by the matrix:
        [2 5]
        [3 7]
        Domain: Vector space of degree 2 and dimension 2 over Rational Field
        User basis matrix:
        [ 1 -1]
        [ 2  3]
        Codomain: Vector space of degree 2 and dimension 2 over Rational Field
        User basis matrix:
        [ 2  1]
        [-1  4]
        sage: psi(u) == 2*x + 5*y
        True
        sage: psi(v) == 3*x + 7*y
        True

    Functions that act on the domain may be used to compute images of
    the domain's basis elements, and this mapping can be extended to
    a unique linear transformation.  The function may be a Python
    function (via ``def`` or ``lambda``) or a Sage symbolic function.  ::

        sage: def g(x):
        ...     return vector(QQ, [2*x[0]+x[2], 5*x[1]])
        ...
        sage: phi = linear_transformation(QQ^3, QQ^2, g)
        sage: phi
        Vector space morphism represented by the matrix:
        [2 0]
        [0 5]
        [1 0]
        Domain: Vector space of dimension 3 over Rational Field
        Codomain: Vector space of dimension 2 over Rational Field

        sage: f = lambda x: vector(QQ, [2*x[0]+x[2], 5*x[1]])
        sage: rho = linear_transformation(QQ^3, QQ^2, f)
        sage: rho
        Vector space morphism represented by the matrix:
        [2 0]
        [0 5]
        [1 0]
        Domain: Vector space of dimension 3 over Rational Field
        Codomain: Vector space of dimension 2 over Rational Field

        sage: x, y, z = var('x y z')
        sage: h(x, y, z) = [2*x + z, 5*y]
        sage: zeta = linear_transformation(QQ^3, QQ^2, h)
        sage: zeta
        Vector space morphism represented by the matrix:
        [2 0]
        [0 5]
        [1 0]
        Domain: Vector space of dimension 3 over Rational Field
        Codomain: Vector space of dimension 2 over Rational Field

        sage: phi == rho
        True
        sage: rho == zeta
        True


    We create a linear transformation relative to non-standard bases,
    and capture its representation relative to standard bases.  With this, we
    can build functions that create the same linear transformation relative
    to the nonstandard bases.  ::

        sage: u = vector(QQ, [1, -1])
        sage: v = vector(QQ, [2, 3])
        sage: D = (QQ^2).subspace_with_basis([u, v])
        sage: x = vector(QQ, [2, 1])
        sage: y = vector(QQ, [-1, 4])
        sage: C = (QQ^2).subspace_with_basis([x, y])
        sage: A = matrix(QQ, [[2, 5], [3, 7]])
        sage: psi = linear_transformation(D, C, A)
        sage: rho = psi.restrict_codomain(QQ^2).restrict_domain(QQ^2)
        sage: rho.matrix()
        [ -4/5  97/5]
        [  1/5 -13/5]

        sage: f = lambda x: vector(QQ, [(-4/5)*x[0] + (1/5)*x[1], (97/5)*x[0] + (-13/5)*x[1]])
        sage: psi = linear_transformation(D, C, f)
        sage: psi.matrix()
        [2 5]
        [3 7]

        sage: s, t = var('s t')
        sage: h(s, t) = [(-4/5)*s + (1/5)*t, (97/5)*s + (-13/5)*t]
        sage: zeta = linear_transformation(D, C, h)
        sage: zeta.matrix()
        [2 5]
        [3 7]

    Finally, we can give an explicit list of images for the basis
    elements of the domain.  ::

        sage: x = polygen(QQ)
        sage: F.<a> = NumberField(x^3+x+1)
        sage: u = vector(F, [1, a, a^2])
        sage: v = vector(F, [a, a^2, 2])
        sage: w = u + v
        sage: D = F^3
        sage: C = F^3
        sage: rho = linear_transformation(D, C, [u, v, w])
        sage: rho.matrix()
        [      1       a     a^2]
        [      a     a^2       2]
        [  a + 1 a^2 + a a^2 + 2]
        sage: C = (F^3).subspace_with_basis([u, v])
        sage: D = (F^3).subspace_with_basis([u, v])
        sage: psi = linear_transformation(C, D, [u+v, u-v])
        sage: psi.matrix()
        [ 1  1]
        [ 1 -1]

    TESTS:

    We test some bad inputs.  First, the wrong things in the wrong places.  ::

        sage: linear_transformation('junk')
        Traceback (most recent call last):
        ...
        TypeError: first argument must be a matrix or a vector space, not junk

        sage: linear_transformation(QQ^2, QQ^3, 'stuff')
        Traceback (most recent call last):
        ...
        TypeError: third argument must be a matrix, function, or list of images, not stuff

        sage: linear_transformation(QQ^2, 'garbage')
        Traceback (most recent call last):
        ...
        TypeError: if first argument is a vector space, then second argument must be a vector space, not garbage

        sage: linear_transformation(QQ^2, Integers(7)^2)
        Traceback (most recent call last):
        ...
        TypeError: vector spaces must have the same field of scalars, not Rational Field and Ring of integers modulo 7

    Matrices must be over a field (or a ring that can be promoted to a field),
    and of the right size.  ::

        sage: linear_transformation(matrix(Integers(6), [[2, 3],[4, 5]]))
        Traceback (most recent call last):
        ...
        TypeError: matrix must have entries from a field, or a ring with a fraction field, not Ring of integers modulo 6

        sage: A = matrix(QQ, 3, 4, range(12))
        sage: linear_transformation(QQ^4, QQ^4, A)
        Traceback (most recent call last):
        ...
        TypeError: domain dimension is incompatible with matrix size

        sage: linear_transformation(QQ^3, QQ^3, A, side='right')
        Traceback (most recent call last):
        ...
        TypeError: domain dimension is incompatible with matrix size

        sage: linear_transformation(QQ^3, QQ^3, A)
        Traceback (most recent call last):
        ...
        TypeError: codomain dimension is incompatible with matrix size

        sage: linear_transformation(QQ^4, QQ^4, A, side='right')
        Traceback (most recent call last):
        ...
        TypeError: codomain dimension is incompatible with matrix size

    Lists of images can be of the wrong number, or not really
    elements of the codomain.  ::

        sage: linear_transformation(QQ^3, QQ^2, [vector(QQ, [1,2])])
        Traceback (most recent call last):
        ...
        ValueError: number of images should equal the size of the domain's basis (=3), not 1

        sage: C = (QQ^2).subspace_with_basis([vector(QQ, [1,1])])
        sage: linear_transformation(QQ^1, C, [vector(QQ, [1,2])])
        Traceback (most recent call last):
        ...
        ArithmeticError: some proposed image is not in the codomain, because
        element (= [1, 2]) is not in free module


    Functions may not apply properly to domain elements,
    or return values outside the codomain.  ::

        sage: f = lambda x: vector(QQ, [x[0], x[4]])
        sage: linear_transformation(QQ^3, QQ^2, f)
        Traceback (most recent call last):
        ...
        ValueError: function cannot be applied properly to some basis element because
        index out of range

        sage: f = lambda x: vector(QQ, [x[0], x[1]])
        sage: C = (QQ^2).span([vector(QQ, [1, 1])])
        sage: linear_transformation(QQ^2, C, f)
        Traceback (most recent call last):
        ...
        ArithmeticError: some image of the function is not in the codomain, because
        element (= [1, 0]) is not in free module

    A Sage symbolic function can come in a variety of forms that are
    not representative of a linear transformation. ::

        sage: x, y = var('x, y')
        sage: f(x, y) = [y, x, y]
        sage: linear_transformation(QQ^3, QQ^3, f)
        Traceback (most recent call last):
        ...
        ValueError: symbolic function has the wrong number of inputs for domain

        sage: linear_transformation(QQ^2, QQ^2, f)
        Traceback (most recent call last):
        ...
        ValueError: symbolic function has the wrong number of outputs for codomain

        sage: x, y = var('x y')
        sage: f(x, y) = [y, x*y]
        sage: linear_transformation(QQ^2, QQ^2, f)
        Traceback (most recent call last):
        ...
        ValueError: symbolic function must be linear in all the inputs:
        unable to convert y to a rational

        sage: x, y = var('x y')
        sage: f(x, y) = [x, 2*y]
        sage: C = (QQ^2).span([vector(QQ, [1, 1])])
        sage: linear_transformation(QQ^2, C, f)
        Traceback (most recent call last):
        ...
        ArithmeticError: some image of the function is not in the codomain, because
        element (= [1, 0]) is not in free module
    """
    from sage.matrix.constructor import matrix
    from sage.modules.module import is_VectorSpace
    from sage.modules.free_module import VectorSpace
    from sage.categories.homset import Hom
    from sage.symbolic.ring import SymbolicRing
    from sage.modules.vector_callable_symbolic_dense import Vector_callable_symbolic_dense
    from inspect import isfunction

    if not side in ['left', 'right']:
        raise ValueError(
            "side must be 'left' or 'right', not {0}".format(side))
    if not (is_Matrix(arg0) or is_VectorSpace(arg0)):
        raise TypeError(
            'first argument must be a matrix or a vector space, not {0}'.
            format(arg0))
    if is_Matrix(arg0):
        R = arg0.base_ring()
        if not R.is_field():
            try:
                R = R.fraction_field()
            except (NotImplementedError, TypeError):
                msg = 'matrix must have entries from a field, or a ring with a fraction field, not {0}'
                raise TypeError(msg.format(R))
        if side == 'right':
            arg0 = arg0.transpose()
            side = 'left'
        arg2 = arg0
        arg0 = VectorSpace(R, arg2.nrows())
        arg1 = VectorSpace(R, arg2.ncols())
    elif is_VectorSpace(arg0):
        if not is_VectorSpace(arg1):
            msg = 'if first argument is a vector space, then second argument must be a vector space, not {0}'
            raise TypeError(msg.format(arg1))
        if arg0.base_ring() != arg1.base_ring():
            msg = 'vector spaces must have the same field of scalars, not {0} and {1}'
            raise TypeError(msg.format(arg0.base_ring(), arg1.base_ring()))

    # Now arg0 = domain D, arg1 = codomain C, and
    #   both are vector spaces with common field of scalars
    #   use these to make a VectorSpaceHomSpace
    # arg2 might be a matrix that began in arg0
    D = arg0
    C = arg1
    H = Hom(D, C, category=None)

    # Examine arg2 as the "rule" for the linear transformation
    # Pass on matrices, Python functions and lists to homspace call
    # Convert symbolic function here, to a matrix
    if is_Matrix(arg2):
        if side == 'right':
            arg2 = arg2.transpose()
    elif isinstance(arg2, (list, tuple)):
        pass
    elif isfunction(arg2):
        pass
    elif isinstance(arg2, Vector_callable_symbolic_dense):
        args = arg2.parent().base_ring()._arguments
        exprs = arg2.change_ring(SymbolicRing())
        m = len(args)
        n = len(exprs)
        if m != D.degree():
            raise ValueError(
                'symbolic function has the wrong number of inputs for domain')
        if n != C.degree():
            raise ValueError(
                'symbolic function has the wrong number of outputs for codomain'
            )
        arg2 = [[e.coeff(a) for e in exprs] for a in args]
        try:
            arg2 = matrix(D.base_ring(), m, n, arg2)
        except TypeError, e:
            msg = 'symbolic function must be linear in all the inputs:\n' + e.args[
                0]
            raise ValueError(msg)
        # have matrix with respect to standard bases, now consider user bases
        images = [v * arg2 for v in D.basis()]
        try:
            arg2 = matrix([C.coordinates(C(a)) for a in images])
        except (ArithmeticError, TypeError), e:
            msg = 'some image of the function is not in the codomain, because\n' + e.args[
                0]
            raise ArithmeticError(msg)
예제 #2
0
def linear_transformation(arg0, arg1=None, arg2=None, side='left'):
    r"""
    Create a linear transformation from a variety of possible inputs.

    FORMATS:

    In the following, ``D`` and ``C`` are vector spaces over
    the same field that are the domain and codomain
    (respectively) of the linear transformation.

    ``side`` is a keyword that is either 'left' or 'right'.
    When a matrix is used to specify a linear transformation,
    as in the first two call formats below, you may specify
    if the function is given by matrix multiplication with
    the vector on the left, or the vector on the right.
    The default is 'left'. Internally representations are
    always carried as the 'left' version, and the default
    text representation is this version.  However, the matrix
    representation may be obtained as either version, no matter
    how it is created.

    - ``linear_transformation(A, side='left')``

      Where ``A`` is a matrix.  The domain and codomain are inferred
      from the dimension of the matrix and the base ring of the matrix.
      The base ring must be a field, or have its fraction field implemented
      in Sage.

    - ``linear_transformation(D, C, A, side='left')``

      ``A`` is a matrix that behaves as above.  However, now the domain
      and codomain are given explicitly. The matrix is checked for
      compatibility with the domain and codomain.  Additionally, the
      domain and codomain may be supplied with alternate ("user") bases
      and the matrix is interpreted as being a representation relative
      to those bases.

    - ``linear_transformation(D, C, f)``

      ``f`` is any function that can be applied to the basis elements of the
      domain and that produces elements of the codomain.  The linear
      transformation returned is the unique linear transformation that
      extends this mapping on the basis elements.  ``f`` may come from a
      function defined by a Python ``def`` statement, or may be defined as a
      ``lambda`` function.

      Alternatively, ``f`` may be specified by a callable symbolic function,
      see the examples below for a demonstration.

    - ``linear_transformation(D, C, images)``

      ``images`` is a list, or tuple, of codomain elements, equal in number
      to the size of the basis of the domain.  Each basis element of the domain
      is mapped to the corresponding element of the ``images`` list, and the
      linear transformation returned is the unique linear transfromation that
      extends this mapping.

    OUTPUT:

    A linear transformation described by the input.  This is a
    "vector space morphism", an object of the class
    :class:`sage.modules.vector_space_morphism`.

    EXAMPLES:

    We can define a linear transformation with just a matrix, understood to
    act on a vector placed on one side or the other.  The field for the
    vector spaces used as domain and codomain is obtained from the base
    ring of the matrix, possibly promoting to a fraction field.  ::

        sage: A = matrix(ZZ, [[1, -1, 4], [2, 0, 5]])
        sage: phi = linear_transformation(A)
        sage: phi
        Vector space morphism represented by the matrix:
        [ 1 -1  4]
        [ 2  0  5]
        Domain: Vector space of dimension 2 over Rational Field
        Codomain: Vector space of dimension 3 over Rational Field
        sage: phi([1/2, 5])
        (21/2, -1/2, 27)

        sage: B = matrix(Integers(7), [[1, 2, 1], [3, 5, 6]])
        sage: rho = linear_transformation(B, side='right')
        sage: rho
        Vector space morphism represented by the matrix:
        [1 3]
        [2 5]
        [1 6]
        Domain: Vector space of dimension 3 over Ring of integers modulo 7
        Codomain: Vector space of dimension 2 over Ring of integers modulo 7
        sage: rho([2, 4, 6])
        (2, 6)

    We can define a linear transformation with a matrix, while explicitly
    giving the domain and codomain.  Matrix entries will be coerced into the
    common field of scalars for the vector spaces.  ::

        sage: D = QQ^3
        sage: C = QQ^2
        sage: A = matrix([[1, 7], [2, -1], [0, 5]])
        sage: A.parent()
        Full MatrixSpace of 3 by 2 dense matrices over Integer Ring
        sage: zeta = linear_transformation(D, C, A)
        sage: zeta.matrix().parent()
        Full MatrixSpace of 3 by 2 dense matrices over Rational Field
        sage: zeta
        Vector space morphism represented by the matrix:
        [ 1  7]
        [ 2 -1]
        [ 0  5]
        Domain: Vector space of dimension 3 over Rational Field
        Codomain: Vector space of dimension 2 over Rational Field

    Matrix representations are relative to the bases for the domain
    and codomain.  ::

        sage: u = vector(QQ, [1, -1])
        sage: v = vector(QQ, [2, 3])
        sage: D = (QQ^2).subspace_with_basis([u, v])
        sage: x = vector(QQ, [2, 1])
        sage: y = vector(QQ, [-1, 4])
        sage: C = (QQ^2).subspace_with_basis([x, y])
        sage: A = matrix(QQ, [[2, 5], [3, 7]])
        sage: psi = linear_transformation(D, C, A)
        sage: psi
        Vector space morphism represented by the matrix:
        [2 5]
        [3 7]
        Domain: Vector space of degree 2 and dimension 2 over Rational Field
        User basis matrix:
        [ 1 -1]
        [ 2  3]
        Codomain: Vector space of degree 2 and dimension 2 over Rational Field
        User basis matrix:
        [ 2  1]
        [-1  4]
        sage: psi(u) == 2*x + 5*y
        True
        sage: psi(v) == 3*x + 7*y
        True

    Functions that act on the domain may be used to compute images of
    the domain's basis elements, and this mapping can be extended to
    a unique linear transformation.  The function may be a Python
    function (via ``def`` or ``lambda``) or a Sage symbolic function.  ::

        sage: def g(x):
        ...     return vector(QQ, [2*x[0]+x[2], 5*x[1]])
        ...
        sage: phi = linear_transformation(QQ^3, QQ^2, g)
        sage: phi
        Vector space morphism represented by the matrix:
        [2 0]
        [0 5]
        [1 0]
        Domain: Vector space of dimension 3 over Rational Field
        Codomain: Vector space of dimension 2 over Rational Field

        sage: f = lambda x: vector(QQ, [2*x[0]+x[2], 5*x[1]])
        sage: rho = linear_transformation(QQ^3, QQ^2, f)
        sage: rho
        Vector space morphism represented by the matrix:
        [2 0]
        [0 5]
        [1 0]
        Domain: Vector space of dimension 3 over Rational Field
        Codomain: Vector space of dimension 2 over Rational Field

        sage: x, y, z = var('x y z')
        sage: h(x, y, z) = [2*x + z, 5*y]
        sage: zeta = linear_transformation(QQ^3, QQ^2, h)
        sage: zeta
        Vector space morphism represented by the matrix:
        [2 0]
        [0 5]
        [1 0]
        Domain: Vector space of dimension 3 over Rational Field
        Codomain: Vector space of dimension 2 over Rational Field

        sage: phi == rho
        True
        sage: rho == zeta
        True


    We create a linear transformation relative to non-standard bases,
    and capture its representation relative to standard bases.  With this, we
    can build functions that create the same linear transformation relative
    to the nonstandard bases.  ::

        sage: u = vector(QQ, [1, -1])
        sage: v = vector(QQ, [2, 3])
        sage: D = (QQ^2).subspace_with_basis([u, v])
        sage: x = vector(QQ, [2, 1])
        sage: y = vector(QQ, [-1, 4])
        sage: C = (QQ^2).subspace_with_basis([x, y])
        sage: A = matrix(QQ, [[2, 5], [3, 7]])
        sage: psi = linear_transformation(D, C, A)
        sage: rho = psi.restrict_codomain(QQ^2).restrict_domain(QQ^2)
        sage: rho.matrix()
        [ -4/5  97/5]
        [  1/5 -13/5]

        sage: f = lambda x: vector(QQ, [(-4/5)*x[0] + (1/5)*x[1], (97/5)*x[0] + (-13/5)*x[1]])
        sage: psi = linear_transformation(D, C, f)
        sage: psi.matrix()
        [2 5]
        [3 7]

        sage: s, t = var('s t')
        sage: h(s, t) = [(-4/5)*s + (1/5)*t, (97/5)*s + (-13/5)*t]
        sage: zeta = linear_transformation(D, C, h)
        sage: zeta.matrix()
        [2 5]
        [3 7]

    Finally, we can give an explicit list of images for the basis
    elements of the domain.  ::

        sage: x = polygen(QQ)
        sage: F.<a> = NumberField(x^3+x+1)
        sage: u = vector(F, [1, a, a^2])
        sage: v = vector(F, [a, a^2, 2])
        sage: w = u + v
        sage: D = F^3
        sage: C = F^3
        sage: rho = linear_transformation(D, C, [u, v, w])
        sage: rho.matrix()
        [      1       a     a^2]
        [      a     a^2       2]
        [  a + 1 a^2 + a a^2 + 2]
        sage: C = (F^3).subspace_with_basis([u, v])
        sage: D = (F^3).subspace_with_basis([u, v])
        sage: psi = linear_transformation(C, D, [u+v, u-v])
        sage: psi.matrix()
        [ 1  1]
        [ 1 -1]

    TESTS:

    We test some bad inputs.  First, the wrong things in the wrong places.  ::

        sage: linear_transformation('junk')
        Traceback (most recent call last):
        ...
        TypeError: first argument must be a matrix or a vector space, not junk

        sage: linear_transformation(QQ^2, QQ^3, 'stuff')
        Traceback (most recent call last):
        ...
        TypeError: third argument must be a matrix, function, or list of images, not stuff

        sage: linear_transformation(QQ^2, 'garbage')
        Traceback (most recent call last):
        ...
        TypeError: if first argument is a vector space, then second argument must be a vector space, not garbage

        sage: linear_transformation(QQ^2, Integers(7)^2)
        Traceback (most recent call last):
        ...
        TypeError: vector spaces must have the same field of scalars, not Rational Field and Ring of integers modulo 7

    Matrices must be over a field (or a ring that can be promoted to a field),
    and of the right size.  ::

        sage: linear_transformation(matrix(Integers(6), [[2, 3],[4, 5]]))
        Traceback (most recent call last):
        ...
        TypeError: matrix must have entries from a field, or a ring with a fraction field, not Ring of integers modulo 6

        sage: A = matrix(QQ, 3, 4, range(12))
        sage: linear_transformation(QQ^4, QQ^4, A)
        Traceback (most recent call last):
        ...
        TypeError: domain dimension is incompatible with matrix size

        sage: linear_transformation(QQ^3, QQ^3, A, side='right')
        Traceback (most recent call last):
        ...
        TypeError: domain dimension is incompatible with matrix size

        sage: linear_transformation(QQ^3, QQ^3, A)
        Traceback (most recent call last):
        ...
        TypeError: codomain dimension is incompatible with matrix size

        sage: linear_transformation(QQ^4, QQ^4, A, side='right')
        Traceback (most recent call last):
        ...
        TypeError: codomain dimension is incompatible with matrix size

    Lists of images can be of the wrong number, or not really
    elements of the codomain.  ::

        sage: linear_transformation(QQ^3, QQ^2, [vector(QQ, [1,2])])
        Traceback (most recent call last):
        ...
        ValueError: number of images should equal the size of the domain's basis (=3), not 1

        sage: C = (QQ^2).subspace_with_basis([vector(QQ, [1,1])])
        sage: linear_transformation(QQ^1, C, [vector(QQ, [1,2])])
        Traceback (most recent call last):
        ...
        ArithmeticError: some proposed image is not in the codomain, because
        element [1, 2] is not in free module


    Functions may not apply properly to domain elements,
    or return values outside the codomain.  ::

        sage: f = lambda x: vector(QQ, [x[0], x[4]])
        sage: linear_transformation(QQ^3, QQ^2, f)
        Traceback (most recent call last):
        ...
        ValueError: function cannot be applied properly to some basis element because
        vector index out of range

        sage: f = lambda x: vector(QQ, [x[0], x[1]])
        sage: C = (QQ^2).span([vector(QQ, [1, 1])])
        sage: linear_transformation(QQ^2, C, f)
        Traceback (most recent call last):
        ...
        ArithmeticError: some image of the function is not in the codomain, because
        element [1, 0] is not in free module

    A Sage symbolic function can come in a variety of forms that are
    not representative of a linear transformation. ::

        sage: x, y = var('x, y')
        sage: f(x, y) = [y, x, y]
        sage: linear_transformation(QQ^3, QQ^3, f)
        Traceback (most recent call last):
        ...
        ValueError: symbolic function has the wrong number of inputs for domain

        sage: linear_transformation(QQ^2, QQ^2, f)
        Traceback (most recent call last):
        ...
        ValueError: symbolic function has the wrong number of outputs for codomain

        sage: x, y = var('x y')
        sage: f(x, y) = [y, x*y]
        sage: linear_transformation(QQ^2, QQ^2, f)
        Traceback (most recent call last):
        ...
        ValueError: symbolic function must be linear in all the inputs:
        unable to convert y to a rational

        sage: x, y = var('x y')
        sage: f(x, y) = [x, 2*y]
        sage: C = (QQ^2).span([vector(QQ, [1, 1])])
        sage: linear_transformation(QQ^2, C, f)
        Traceback (most recent call last):
        ...
        ArithmeticError: some image of the function is not in the codomain, because
        element [1, 0] is not in free module
    """
    from sage.matrix.constructor import matrix
    from sage.modules.module import is_VectorSpace
    from sage.modules.free_module import VectorSpace
    from sage.categories.homset import Hom
    from sage.symbolic.ring import SymbolicRing
    from sage.modules.vector_callable_symbolic_dense import Vector_callable_symbolic_dense
    from inspect import isfunction

    if not side in ['left', 'right']:
        raise ValueError("side must be 'left' or 'right', not {0}".format(side))
    if not (is_Matrix(arg0) or is_VectorSpace(arg0)):
        raise TypeError('first argument must be a matrix or a vector space, not {0}'.format(arg0))
    if is_Matrix(arg0):
        R = arg0.base_ring()
        if not R.is_field():
            try:
                R = R.fraction_field()
            except (NotImplementedError, TypeError):
                msg = 'matrix must have entries from a field, or a ring with a fraction field, not {0}'
                raise TypeError(msg.format(R))
        if side == 'right':
            arg0 = arg0.transpose()
            side = 'left'
        arg2 = arg0
        arg0 = VectorSpace(R, arg2.nrows())
        arg1 = VectorSpace(R, arg2.ncols())
    elif is_VectorSpace(arg0):
        if not is_VectorSpace(arg1):
            msg = 'if first argument is a vector space, then second argument must be a vector space, not {0}'
            raise TypeError(msg.format(arg1))
        if arg0.base_ring() != arg1.base_ring():
            msg = 'vector spaces must have the same field of scalars, not {0} and {1}'
            raise TypeError(msg.format(arg0.base_ring(), arg1.base_ring()))

    # Now arg0 = domain D, arg1 = codomain C, and
    #   both are vector spaces with common field of scalars
    #   use these to make a VectorSpaceHomSpace
    # arg2 might be a matrix that began in arg0
    D = arg0
    C = arg1
    H = Hom(D, C, category=None)

    # Examine arg2 as the "rule" for the linear transformation
    # Pass on matrices, Python functions and lists to homspace call
    # Convert symbolic function here, to a matrix
    if is_Matrix(arg2):
        if side == 'right':
            arg2 = arg2.transpose()
    elif isinstance(arg2, (list, tuple)):
        pass
    elif isfunction(arg2):
        pass
    elif isinstance(arg2, Vector_callable_symbolic_dense):
        args = arg2.parent().base_ring()._arguments
        exprs = arg2.change_ring(SymbolicRing())
        m = len(args)
        n = len(exprs)
        if m != D.degree():
            raise ValueError('symbolic function has the wrong number of inputs for domain')
        if n != C.degree():
            raise ValueError('symbolic function has the wrong number of outputs for codomain')
        arg2 = [[e.coefficient(a) for e in exprs] for a in args]
        try:
            arg2 = matrix(D.base_ring(), m, n, arg2)
        except TypeError as e:
            msg = 'symbolic function must be linear in all the inputs:\n' + e.args[0]
            raise ValueError(msg)
        # have matrix with respect to standard bases, now consider user bases
        images = [v*arg2 for v in D.basis()]
        try:
            arg2 = matrix([C.coordinates(C(a)) for a in images])
        except (ArithmeticError, TypeError) as e:
            msg = 'some image of the function is not in the codomain, because\n' + e.args[0]
            raise ArithmeticError(msg)
    else:
        msg = 'third argument must be a matrix, function, or list of images, not {0}'
        raise TypeError(msg.format(arg2))

    # arg2 now compatible with homspace H call method
    # __init__ will check matrix sizes versus domain/codomain dimensions
    return H(arg2)