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Then the two compositions are
BA = 0 −1 1 0 = 0 1
1 0 0 −1 1 0
Algebra of linear transformations and
AB = 1 0 0 −1 = 0 −1
matrices 0 −1 1 0 −1 0
Math 130 Linear Algebra
DJoyce, Fall 2013 The products aren’t the same.
You can perform these on physical objects. Take
We’ve looked at the operations of addition and ◦
a book. First rotate it 90 then flip it over. Start
scalar multiplication on linear transformations and again but flip first then rotate 90◦. The book ends
used them to define addition and scalar multipli- up in different orientations.
cation on matrices. For a given basis β on V and
another basis γ on W, we have an isomorphism Matrix multiplication is associative. Al-
γ ≃
φ : Hom(V,W) → M of vector spaces which
β m×n though it’s not commutative, it is associative.
assigns to a linear transformation T : V → W its That’s because it corresponds to composition of
standard matrix [T]γ.
β functions, and that’s associative. Given any three
Wealso have matrix multiplication which corre- functions f, g, and h, we’ll show (f ◦ g) ◦ h =
sponds to composition of linear transformations. If f ◦ (g ◦ h) by showing the two sides have the same
A is the standard matrix for a transformation S, values for all x.
and B is the standard matrix for a transformation
T, then we defined multiplication of matrices so ((f ◦ g) ◦ h)(x) = (f ◦ g)(h(x)) = f(g(h(x)))
that the product AB is be the standard matrix for
S◦T. while
There are a few more things we should look at (f ◦ (g ◦ h))(x) = f((g ◦ h)(x)) = f(g(h(x))).
for matrix multiplication. It’s not commutative.
It is associative. It distributes with matrix addi- They’re the same.
tion. There are identity matrices I for multiplica- Since composition of functions is associative, and
tion. Cancellation doesn’t work. You can compute linear transformations are special kinds of func-
powers of square matrices. And scalar matrices. tions, therefore composition of linear transforma-
tions is associative. Since matrix multiplication
Matrix multiplication is not commutative. corresponds to composition of linear transforma-
It shouldn’t be. It corresponds to composition of tions, therefore matrix multiplication is associative.
linear transformations, and composition of func- An alternative proof would actually involve
tions is not commutative. computations, probably with summation notation,
Example 1. Let’s take a 2-dimensional geometric something like
example. Let T be rotation 90◦ clockwise, and S be !
reflection across the x-axis. We’ve looked at those Xa Xb c
ij jk kl
before. The standard matrices A for S and B for j k
T are = Xa b c
ij jk kl
A = 1 0 j,k !
0 −1 X X
= a b c .
0 −1 ij jk kl
B = 1 0 k j
1
Matrix multiplication distributes over ma- AI = A = IA are different. For example,
trix addition. When A, B, and C are the right 4 5 6 1 0 0
shape matrices so the the operations can be per- 0 1 0
formed, then the the following are always identities: 3 −1 0 0 0 1
A(B+C) = AB+AC = 4 5 6
(A+B)C = AC+BC 3 −1 0
= 1 0 4 5 6 .
Whydoesitwork? It suffices to show that it works 0 1 3 −1 0
for linear transformations. Suppose that R, S, and
T are their linear transformations. The correspond- Cancellation doesn’t work for matrix multi-
ing identities are plication! Notonlyis matrix multiplication non-
commutative, but the cancellation law doesn’t hold
R◦(S+T) = (R◦S)+(R◦T) for it. You’re familiar with cancellation for num-
(R+S)◦T = (R◦T)+(S◦T) bers: if xy = xz but x 6= 0, then y = z. But we
can come up with matrices so that AB = AC and
Simply evaluate them at a vector v and see that 1 0
A 6= 0, but B 6= C. For example A = 0 0 ,
you get the same thing. Here’s the first identity. 1 0 1 0
You’ll need to use linearity of R at one point. B= , and C = .
0 3 0 4
(R◦(S+T))(v) = R((S+T)(v) Powers of matrices. Frequently, we’ll multiply
= R(S(v+T(v) square matrices by themselves (you can only mul-
= R(S(v))+R(T(v)) tiply square matrices by themselves), and we’ll use
((R◦S)+(R◦T))(v) = (R◦S)(v)+(R◦T)(v) the standard notation for powers. The expression
= R(S(v))+R(T(v)) Ap stands for the product of p copies of A. Since
matrix multiplication is associative, this definition
works, so long as p is a positive integer. But we can
The identity matrices. Just like there are ma- extend the definition to p = 0 by making A0 = I,
trices that work as additive identities (we denoted and the usual properties will will still hold. That is,
them all 0 as described above), there are matrices p q p+q p q pq
A A =A and (A ) = A . Later, we’ll extend
that work as multiplicative identities, and we’ll de- powers to the case when A is an invertible matrix
note them all I and all them identity matrices. An and the power p is a negative integer.
identity matrix is a square n by n matrix with 1 Warning: because matrix multiplication is not
down the diagonal and 0 elsewhere. You could de- commutative in general, it is usually the case that
note them In to emphasize their sizes, but you can p p p
always tell by the context what its size is, so we’ll (AB) 6= A B .
leave out the index n. By the way, whenever you’ve Scalar matrices. A scalar matrix is a matrix
got a square n by n matrix, you can say the order with the scalar r down the diagonal. That’s the
of the matrix is n. Anyway, I acts like an identity same thing as the scalar r times the identity ma-
matrix trix. For instance,
AI =A=IA.
4 0 0 1 0 0
Note that if A is not a square matrix, then the
0 4 0 =4 0 1 0 =4I.
orders of the two identity matrices I in the identity 0 0 4 0 0 1
2
Among other things, that means that we can iden-
tify a scalar matrix with the scalar.
Math 130 Home Page at
http://math.clarku.edu/~djoyce/ma130/
3
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