Page 206 - A Course in Linear Algebra with Applications

190 Chapter Six: Linear Transformations

More unexpectedly, when we compose the linear opera-
tions Ti and T 2, the resulting linear operator T1T2 is repre-
sented by the product of the matrices representing T\ and T 2 •
In technical language, the function which sends T to
M(T) is an algebra isomorphism from L(V) to M n(F). The
main point here is that isomorphic algebras, like isomorphic
vector spaces, are to be regarded as similar objects, which
exhibit the same essential features, even although their un-
derlying sets may be quite different.
In conclusion, our vague feeling that the algebras L(V)
and M n(F) are somehow quite closely related is made precise
by the assertion that the algebra of all linear operators on an
n- dimensional vector space over a field F is isomorphic with
the algebra of all n x n matrices over F.

Example 6.3.5
Prove part (iii) of Theorem 6.3.10.

Let v be any vector of the vector space; then, using the
fundamental equation [T(v)]s = M(T)[v]s, we obtain

[TiT 2{-v)} B = M(T 1 )[r 2 (v)] s = M(T 1 )(M(T 2 )[v] s )
= M(7i)M(r 2 )[v] s ,

which shows that M{T XT 2) = M(Ti)M(T 2), as required.

Exercises 6.3

1. Find bases for the kernel and image of the following linear
transformations:
(a) T : R 4 —>• R where T sends a column to the sum
of its entries;
(b) T : P 3 (R) - P 3 (R) where T(f) = ';
/
< = ) T : R ^ R ° where r ( ( * ) ) = ( £ ; $ ) .