Linear Algebra, 2020a

236 Chapter Three. Maps Between Spaces
IV.2
Matrix Multiplication
After representing addition and scalar multiplication of linear maps in the prior
subsection, the natural next operation to consider is function composition.
2.1 Lemma The composition of linear maps is linear.
Proof (Note: this argument has already appeared, as part of the proof of
Theorem I.2.2.) Let h: V → W and g: W → U be linear. The calculation
g ◦ h ( ) (
c 1 · ⃗v 1 + c 2 · ⃗v 2 = g h(c1 · ⃗v 1 + c 2 · ⃗v 2 ) ) = g ( c 1 · h(⃗v 1 )+c 2 · h(⃗v 2 ) )
= c 1 · g ( h(⃗v 1 )) + c 2 · g(h(⃗v 2 ) ) = c 1 · (g ◦ h)(⃗v 1 )+c 2 · (g ◦ h)(⃗v 2 )
shows that g ◦ h: V → U preserves linear combinations, and so is linear. QED
As we did with the operation of matrix addition and scalar multiplication,
we will see how the representation of the composite relates to the representations
of the compositors by first considering an example.
2.2 Example Let h: R 4 → R 2 and g: R 2 → R 3 , fix bases B ⊂ R 4 , C ⊂ R 2 ,
D ⊂ R 3 , and let these be the representations.
⎛ ⎞
H = Rep B,C (h) =
(
4 6 8 2
5 7 9 3
)
B,C
⎜
1 1 ⎟
G = Rep C,D (g) = ⎝0 1⎠
1 0
To represent the composition g ◦ h: R 4 → R 3 we start with a ⃗v, represent h of
⃗v, and then represent g of that. The representation of h(⃗v) is the product of h’s
matrix and ⃗v’s vector.
(
)
4 6 8 2
Rep C ( h(⃗v))=
5 7 9 3
B,C
⎛ ⎞
v 1
v 2
⎜ ⎟
⎝v 3 ⎠
v 4
B
=
C,D
(
)
4v 1 + 6v 2 + 8v 3 + 2v 4
5v 1 + 7v 2 + 9v 3 + 3v 4
The representation of g( h(⃗v)) is the product of g’s matrix and h(⃗v)’s vector.
⎛
⎜
1 1
⎞
(
)
⎟ 4v 1 + 6v 2 + 8v 3 + 2v 4
Rep D ( g(h(⃗v)) )= ⎝0 1⎠
5v 1 + 7v 2 + 9v 3 + 3v 4
1 0
C
C,D
⎛
⎜
1 · (4v 1 + 6v 2 + 8v 3 + 2v 4 )+1 · (5v 1 + 7v 2 + 9v 3 + 3v 4 )
⎞
⎟
= ⎝0 · (4v 1 + 6v 2 + 8v 3 + 2v 4 )+1 · (5v 1 + 7v 2 + 9v 3 + 3v 4 ) ⎠
1 · (4v 1 + 6v 2 + 8v 3 + 2v 4 )+0 · (5v 1 + 7v 2 + 9v 3 + 3v 4 )
C
D