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An interpretation of the product of two matrices

Suppose that the matrix A carries the vector X1 into the vector Y1, the vector X2 into the vector Y2 , etc. i.e.

AX1 = Y1

AX2 = Y2

......

......

AXn = Yn

Then

A [ X1 X2 .... Xn] = [Y1 Y2 ... Yn]

where [ X1 X2 .... Xn] is a matrix whose columns are X1, X2, .... , Xn and [Y1 Y2 ... Yn] is a matrix whose columns are Y1,Y2, ... ,Yn .

The above stated fact provides us with a way of viewing the product of one matrix by another. In the product

AB = C

the columns of B can be viewed as a set of vectors which are carried by the matrix A into the columns of C (i.e. the matrix A is viewed as an operator that operates on the columns of B, viewed as vectors, and carries them into the columns of C, also viewed as vectors).

In the same way B can be viewed as the operator matrix and the rows of A viewed as vectors which are carried into the rows of C (viewed as vectors).