5.1. MATRIX ARITHMETIC 85

is the m×1 column vector which equals the following linear combination of the columns.

v1a1 + v2a2 + · · ·+ vnan ≡n

∑j=1

v ja j (5.9)

If the jth column of A is A1 j

A2 j...

Am j

then 5.9 takes the form

v1

A11

A21...

Am1

+ v2

A12

A22...

Am2

+ · · ·+ vn

A1n

A2n...

Amn

Thus the ith entry of Av is ∑

nj=1 Ai jv j. Note that multiplication by an m×n matrix takes an

n×1 matrix, and produces an m×1 matrix (vector).

Here is another example.

Example 5.1.10 Compute

 1 2 1 30 2 1 −22 1 4 1



1201

 .

First of all this is of the form (3×4)(4×1) and so the result should be a (3×1) . Notehow the inside numbers cancel. To get the element in the second row and first and onlycolumn, compute

4

∑k=1

a2kvk = a21v1 +a22v2 +a23v3 +a24v4

= 0×1+2×2+1×0+(−2)×1 = 2.

You should do the rest of the problem and verify

 1 2 1 30 2 1 −22 1 4 1



1201

=

 825

 .

The next task is to multiply an m×n matrix times an n× p matrix. Before doing so, thefollowing may be helpful.