"""
There is an explicit isomorphism between all finite-dimensional vector
spaces. In particular, there is an isomorphism between the m-by-n
matrices and `$R^(m \times n)$`. Since most vector operations are not
available on Sage matrices, we have to go back and forth between these
two vector spaces often.
"""

from sage.all import *

def isomorphism(matrix_space):
    """
    Create isomorphism (i.e. the function) that converts elements
    of a matrix space into those of the corresponding finite-dimensional
    vector space.

    INPUT:

    - matrix_space: A finite-dimensional ``MatrixSpace`` object.

    OUTPUT:

    - (phi, phi_inverse): If ``matrix_space`` has dimension m*n, then
                          ``phi`` will map m-by-n matrices to R^(m*n).
                          The inverse mapping ``phi_inverse`` will go
                          the other way.

    EXAMPLES:

        sage: M = MatrixSpace(QQ,4,4)
        sage: (p, p_inv) = isomorphism(M)
        sage: m = M(range(0,16))
        sage: p_inv(p(m)) == m
        True

    """
    from sage.matrix.matrix_space import is_MatrixSpace
    if not is_MatrixSpace(matrix_space):
        raise TypeError('argument must be a matrix space')

    base_ring = matrix_space.base_ring()
    vector_space = VectorSpace(base_ring, matrix_space.dimension())

    def phi(m):
        return vector_space(m.list())

    def phi_inverse(v):
        return matrix_space(v.list())

    return (phi, phi_inverse)