A First Course in Linear Algebra · Problem CRS.M21

Matrices — Robert Beezer

Mathematics — Linear Algebra · column-space-rref-procedure

Problem

We have a variety of theorems about how to create column spaces and row spaces and they frequently involve row-reducing a matrix. Here is a procedure that some try to use to get a column space. Begin with an m×nm \times n matrix AA and row-reduce to a matrix BB with columns B1,B2,B3,…,Bn\mathbf{B}_1, \mathbf{B}_2, \mathbf{B}_3, \ldots, \mathbf{B}_n. Then form the column space of AA as C(A)=⟨{B1,B2,B3,…,Bn}⟩=C(B)\mathcal{C}(A) = \langle \{\mathbf{B}_1, \mathbf{B}_2, \mathbf{B}_3, \ldots, \mathbf{B}_n\} \rangle = \mathcal{C}(B) This is not a legitimate procedure, and therefore is not a theorem. Construct an example to show that the procedure will not in general create the column space of AA.

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