Page 183 - A Course in Linear Algebra with Applications

6.2: Linear Transformations and Matrices 167

Now consider the effect of T on an arbitrary vector of V,
say v = C1V1 + • • • + c n v n . This is computed by using the
expression for T(VJ) given above:

n n n m

3 = 1 3 = 1 3 = 1 i=l

On interchanging the order of summations, this becomes

m n
T V a c w
( ) = ^ E V j ) i -
i=l j=l

Hence the coordinate vector of T(v) with respect to the or-
a c
2
dered basis C has entries J2^=i ij j for i = 1, ,..., m. This
means that
= A[v] B.
[T(v)] c
The conclusions of this discussion can be summed up as
follows.
Theorem 6.2.3
Let T : V —> W be a linear transformation between two non-
zero finite-dimensional vector spaces V and W over the same
field. Suppose that B and C are ordered bases for V and W
respectively. If v is any vector of V, then

=
[T(v)] c A[v] B
where A is the mxn matrix whose jth column is the coordinate
vector of the image under T of the jth vector of B, taken with
respect to the basis C.

What this result means is that a linear transformation
between non-zero finite-dimensional vector spaces can always
be represented by left multiplication by a suitable matrix. At
this point the reader may wonder if it is worth the trouble