378 CHAPTER 18. LINEAR FUNCTIONS

Example 18.1.14(

1 2 34 −5 −8

)+

(1 2 34 5 6

)=

(2 4 68 0 −2

).

Example 18.1.15 Find(

1 2 34 −5 −8

) 2 81 02 −2

+

(1 22 1

)

(1 2 34 −5 −8

) 2 81 02 −2

+

(1 22 1

)

=

(10 2−13 48

)+

(1 22 1

)=

(11 4−11 49

)Although matrix multiplication (composition of linear transformations) is not commu-

tative, it does have several very important properties.

Proposition 18.1.16 If all multiplications and additions make sense, the followinghold for matrices A,B,C and a,b scalars.

A(aB+bC) = a(AB)+b(AC) (18.10)

(B+C)A = BA+CA (18.11)

A(BC) = (AB)C (18.12)

Proof: Using the definition for matrix multiplication, (A(aB+bC))i j =

∑k

Aik (aB+bC)k j = ∑k

Aik(aBk j +bCk j

)= a∑

kAikBk j +b∑

kAikCk j

= a(AB)i j +b(AC)i j = (a(AB)+b(AC))i j .

Thus A(B+C) = AB+AC as claimed. Formula 18.11 is entirely similar.Formula 18.12 is the associative law of multiplication. Using Definition 18.1.11,

(A(BC))i j = ∑k

Aik (BC)k j = ∑k

Aik ∑l

BklCl j

= ∑l(AB)il Cl j = ((AB)C)i j .

This proves 18.12.Specializing 18.10 to the case where B,C are vectors, this shows that x→ Ax is a linear

transformation. Thus every linear transformation can be realized by matrix multiplicationand conversely, if you consider matrix multiplication, this is a linear transformation. Thisis why in this book, I will emphasize matrix multiplication rather than the abstract conceptof a linear transformation.

Also note that 18.12, along with the theorem that matrix multiplication corresponds tocomposition of linear transformations, follows from the general observation from collegealgebra that

S◦ (T ◦V ) = (S◦T )◦V

As to the restriction 18.1, it is essentially the statement that if you want S ◦ T, then thepossible values of T must be in the domain of S.