Page 198 - A Course in Linear Algebra with Applications

182 Chapter Six: Linear Transformations

Theorem 6.3.4
A linear transformation T : V —> W is an isomorphism if and
only if Ker(T) is the zero subspace of V and Im(T) equals
W. Moreover, if T is an isomorphism, then so is its inverse
T~ l : W -> V.

Proof
The first statement follows from 6.3.2. As for the second state-
l
ment, all that need be shown is that T~ is actually a linear
transformation: for by 6.1.5 it certainly has an inverse. This
is achieved by a trick. Let vj and v 2 be any two vectors in V.
Then certainly

1
T ( r - ( v 1 + v 2 ) ) = v 1 + v 2 ,
while on the other hand,

X
TOT-VI) + T- M) = nr-^vi)) + rcr-Va))
= vi + v 2 ,
because T is known to be a linear transformation. Since T
is an injective function, this can only mean that the vectors
- 1
- 1
T (vi + v 2 ) and T (vi) +T _ 1 (v 2 ) are equal; for they have
the same image under T.
_ 1
In a similar way it can be demonstrated that T (cvi)
_ 1
c
equals T (vi) where c is any scalar: just check that both
sides have the same image under T. Hence T~ l is a linear
transformation.
Two vector spaces V and W are said to be isomorphic if
there is an isomorphism from one to the other. Observe that
isomorphic vector spaces are necessarily over the same field of
scalars. The notation
V ~W

is often used to express the fact that vector spaces V and W
are isomorphic.