Column space

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The column space of an m-by-n matrix with real entries is the subspace of R^m 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 R^m, 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, .... , a_n], then Col Z = Span {a₁, ...., a_n}