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A vector_space is a set of vectors such that (1) if we add any two
vectors x
and y
in the space, their sum x+y
is also
in the space, and (2) if we multiply any vector x
in the space by
a scalar c
, then cx
is still in the space. The
vector_space
class is built using the matrix
(see section Matrices for Linear Algebra) class and the
fract_vector_list class (a list of fract_vectors
). The
vector space is represented either by a list independent vectors, or by
the column vectors of the matrix. Only one of the two representations
is stored at any given time. If the other representation is needed,
then it is recalculated. The class also contains an integer
space
that corresponds to the dimension of vector space; that is,
the maximum number of dimensions that this vector space could possibly
span. All basis vectors in the fract_vector_list
representation
must have space
elements. Alternatively, the matrix in the
matrix
representation must have space
rows.
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