16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 1.5.6

Systems of Linear Equations

Mathematics — Linear Algebra · matrix-multiplication-columns

Problem

Let

A

be an

m \times n

matrix and

B

n \times k

matrix. Show that the columns of

C = AB

are linear combinations of the columns of

A

. If

\alpha_1, \ldots, \alpha_n

are the columns of

A

and

\gamma_1, \ldots, \gamma_k

are the columns of

C

, then

\gamma_j = \sum_{r=1}^{n} B_{rj} \alpha_r.