Page 47 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

(9)

Remark Formulas 8 and 9 are special cases of a more general procedure for multiplying partitioned matrices (see Exercises
15, 16 and 17).

Matrix Products as Linear Combinations

Row and column matrices provide an alternative way of thinking about matrix multiplication. For example, suppose that

Then

(10)

In words, 10 tells us that the product of a matrix A with a column matrix x is a linear combination of the column matrices

of A with the coefficients coming from the matrix x. In the exercises we ask the reader to show that the product of a

matrix y with an matrix A is a linear combination of the row matrices of A with scalar coefficients coming from y.

EXAMPLE 8 Linear Combinations
The matrix product

can be written as the linear combination of column matrices

The matrix product

can be written as the linear combination of row matrices

It follows from 8 and 10 that the jth column matrix of a product is a linear combination of the column matrices of A with
the coefficients coming from the jth column of B.