Page 202 - A Course in Linear Algebra with Applications

186 Chapter Six: Linear Transformations

Proof
Since isomorphic vector spaces have the same dimension, we
have dim((C7 + W)/W) = dxm(U/(U D W)). Now use the
formula for the dimension of a quotient space in 5.3.7 to obtain

dim(U + W) - dim(W) = dim(U) - dim(U D W),

from which the result follows.

The algebra of linear operators on a vector space
We conclude the chapter by observing that the set of all
linear operators on a vector space has certain formal properties
which are very similar to properties that have already been
seen to hold for matrices. This similarity can be expressed by
saying that both systems form what is called an algebra.
Consider a vector space V with finite dimension n over a
field F. Let Ti and Ti be two linear operators on V. Then
we define their sum Ti + T2 by the rule

r 1 + T 2 ( v ) = Ti(v)+T 2 (v)

and also the scalar multiple cTi, where c is an element of F,
by
cTi(v)=c(Ti(v)).

It is quite routine to verify that T\ + T 2 and cT\ are also linear
operators on V. For example, to show that T\ + T 2 is a linear
operator we compute

Ti + T 2 (v x + v 2 ) = Ti (vi + v 2 ) + T 2 (vi + v 2 )
= Ti(vi) + Ti(v 2 ) + T 2 ( Vl ) + T 2 (v 2 ),

from which it follows that

Ti +T 2 ( V l + v 2 ) = (Ti +r 2 (vi)) + (Ti +T 2 (v 2 )).