Linear Algebra, 2020a

Section IV. Matrix Operations 241
example, for associativity the i, j entry of (FG)H is
(f i,1 g 1,1 + f i,2 g 2,1 + ···+ f i,r g r,1 )h 1,j
+(f i,1 g 1,2 + f i,2 g 2,2 + ···+ f i,r g r,2 )h 2,j
.
+(f i,1 g 1,s + f i,2 g 2,s + ···+ f i,r g r,s )h s,j
where F, G, and H are m×r, r×s, and s×n matrices. Distribute
f i,1 g 1,1 h 1,j + f i,2 g 2,1 h 1,j + ···+ f i,r g r,1 h 1,j
and regroup around the f’s
+ f i,1 g 1,2 h 2,j + f i,2 g 2,2 h 2,j + ···+ f i,r g r,2 h 2,j
.
.
+ f i,1 g 1,s h s,j + f i,2 g 2,s h s,j + ···+ f i,r g r,s h s,j
f i,1 (g 1,1 h 1,j + g 1,2 h 2,j + ···+ g 1,s h s,j )
+ f i,2 (g 2,1 h 1,j + g 2,2 h 2,j + ···+ g 2,s h s,j )
.
+ f i,r (g r,1 h 1,j + g r,2 h 2,j + ···+ g r,s h s,j )
to get the i, j entry of F(GH).
Contrast the two proofs. The index-heavy argument is hard to understand in
that while the calculations are easy to check, the arithmetic seems unconnected
to any idea. The argument in the proof is shorter and also says why this property
“really” holds. This illustrates the comments made at the start of the chapter on
vector spaces — at least sometimes an argument from higher-level constructs is
clearer.
We have now seen how to represent the composition of linear maps. The
next subsection will continue to explore this operation.
Exercises
̌ 2.14 Compute, or state “not defined”.
( )( ) ( ) ⎛ ⎞
−1 −1
3 1 0 5
1 1 −1
(a)
(b)
⎝2
3 1 1 ⎠
−4 2 0 0.5 4 0 3
3 1 1
( ) ⎛ ⎞
1 0 5 ( )( )
2 −7
(c)
⎝−1 1 1⎠
5 2 −1 2
(d)
7 4
3 1 3 −5
3 8 4