Linear Algebra Done Right · Problem 3C.9

Linear Maps — Sheldon Axler

Mathematics — Linear Algebra · matrices-row-column

Problem

Suppose

a = \begin{pmatrix} a_1 & \cdots & a_n \end{pmatrix}

is a 1-by-

n

matrix and

B

is an

n

-by-

p

matrix. Prove that

aB = a_1 B_{1,\cdot} + \cdots + a_n B_{n,\cdot}.

In other words, show that

aB

is a linear combination of the rows of

B

, with the scalars that multiply the rows coming from

a

. This exercise gives the row version of 3.50.