18.1. THE MATRIX OF A LINEAR TRANSFORMATION 375

Proof: This follows from the above definition and Proposition 18.1.3.

Example 18.1.7 Write the following as a matrix times a vector.

2

 111

−2

 123

+3

 0−42

According to the above definition, this is of the form 1 1 0

1 2 −41 3 2

 2−23

When you multiply a matrix times a vector, you are just specifying a linear combination

of the columns of the matrix. Thus every linear function T : Rn → Rm can be written asfollows:

Tx= Ax

where A is an m× n matrix called the matrix of the linear transformation. This matrix isdenoted as [T ]. This is formalized in the following definition.

Definition 18.1.8 Let A be an m×n matrix. Then Ai j will denote the number in theith row and jth column. [T ] denotes the m×n matrix such that T (x) = [T ]x.

Example 18.1.9 Say A =

(1 2 −54 −7 2

). Then A11 = 1,A12 = 2,A23 = 2,A22 = −7

etc.

When writing Ai j the first index i always refers to the row and the second listed indexrefers to the column. This is hard for some of us to remember. Perhaps it will help tothink Rowman Catholic. Another thing which is sometimes hard to remember is that thecolumns are vertical like those on the Parthenon in Athens and the rows are horizontal likethe rows made by a tractor pulling a plow.

Definition 18.1.10 Suppose T : Rn → Rm and S : Rm → Rp. You could considerthe composition of these functions S◦T defined as S◦T (x)≡ S (T (x)) .

With this definition, which is really nothing more than a re-statement of definitions frompre-calculus or algebra, the following is a fundamental theorem. It says that, appropriatelydefined, matrix multiplication corresponds to composition of linear transformations. Thisdefinition will be as follows.

Definition 18.1.11 Let A be an m×n matrix and let B be an n× p matrix. Then

(AB)i j =n

∑k=1

AikBk j = Ai1B1 j +Ai2B2 j + · · ·+AinBn j

In terms of familiar concepts, the i jth entry of AB is the ith row of A times the jth column ofB meaning you take the dot product of the ith row of A with the jth column of B. Note thatAx is a special case of this. Indeed,

(Ax)i = ∑k

Aikxk