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Row and column spaces
Vector spaces associated to a matrix / From Wikipedia, the free encyclopedia
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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.
![Thumb image](http://upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Matrix_Rows.svg/320px-Matrix_Rows.svg.png)
![Thumb image](http://upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Matrix_Columns.svg/320px-Matrix_Columns.svg.png)
Let be a field. The column space of an m × n matrix with components from
is a linear subspace of the m-space
. The dimension of the column space is called the rank of the matrix and is at most min(m, n).[1] A definition for matrices over a ring
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.[2]
This article considers matrices of real numbers. The row and column spaces are subspaces of the real spaces and
respectively.[3]