Page 179 - A Course in Linear Algebra with Applications

6.2: Linear Transformations and Matrices 163

where F is some field of scalars. Let {Ei,E2, ...,E n} be the
standard basis of F n written in the usual order, that of the
columns of the identity matrix l n . Also let {Di,D2, ...,D m}
m
be the corresponding ordered basis of F . Since T(Ej) is a
m
vector in F , it can be written in the form

/ O i j
T{E 3) = — aijDi + • • • + a mjD m = 2_^ &ijDi

\ Uj m3
Put A = [ajj] m)n , so that the columns of the matrix A are
the vectors T(E{), ...,T(E n). We show that T is completely
determined by the matrix A.
n
Take an arbitrary vector in F , say

x E
X = : J = xiE 1 + \~x nE n = Y^ j r
\x nl i=i

Then, using 6.2.1 together with the expression for T(Ej), we
obtain
n n n m

j=l j=l j-l i=l
m n
= a x D
52(52 i3 j) i-
i=l j = l

Therefore the ith entry of T(X) equals the ith entry of the
matrix product AX. Thus we have shown that

T{X) = AX,

n
which means that the effect of T on a vector in F is to
multiply it on the left by the matrix A. Thus A determines T
completely.