Column space

In linear algebra, the column space (also called the range or image) of a matrix A is the span (set of all possible linear combinations) of its column vectors. The column space of a matrix is the image or range of the corresponding matrix transformation.
Let




F



{\displaystyle \mathbb {F} }
be a field. The column space of an m × n matrix with components from




F



{\displaystyle \mathbb {F} }
is a linear subspace of the m-space





F


m




{\displaystyle \mathbb {F} ^{m}}
. The dimension of the column space is called the rank of the matrix and is at most min(m, n). A definition for matrices over a ring




K



{\displaystyle \mathbb {K} }
is also possible.
The row space is defined similarly.
The row space and the column space of a matrix A are sometimes denoted as C(AT) and C(A) respectively.This article considers matrices of real numbers. The row and column spaces are subspaces of the real spaces





R


n




{\displaystyle \mathbb {R} ^{n}}
and





R


m




{\displaystyle \mathbb {R} ^{m}}
respectively.

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