Column Space

The vector space generated by the columns of a matrix viewed as vectors. The column space of an n×m matrix A with real entries is a subspace generated by m elements of R^n, hence its dimension is at most min(m,n). It is equal to the dimension of the row space of A and is called the rank of A.

The matrix A is associated with a linear transformation T:R^m->R^n, defined by


for all vectors x of R^m, which we suppose written as column vectors. Note that Ax is the product of an n×m and an m×1 matrix, hence it is an n×1 matrix according to the rules of matrix multiplication. In this framework, the column vectors of A are the vectors T(e_1),...,T(e_m), where e_1,...,e_m are the elements of the standard basis of R^m. This shows that the column space of A is the range of T, and explains why the dimension of the latter is equal to the rank of A.

See also

Row Space

This entry contributed by Margherita Barile

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Cite this as:

Barile, Margherita. "Column Space." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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