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Column space

The column space of an m-by-n matrix with real entries is the subspace of Rm generated by the column vectors of the matrix. Its dimension is the rank of the matrix and is at most min(m,n).

If one considers the matrix as a linear transformation from Rn to Rm, then the column space of the matrix equals the image of this linear transformation.

The column spaces of a matrix Z is the set of all linear combinations of the columns in Z. If Z = [a1, .... , an], then Col Z = Span {a1, ...., an}

See also row space.