Page 199 - A Course in Linear Algebra with Applications

6.3: Kernel, Image and Isomorphism 183

How can one tell if two finite-dimensional vector spaces
are isomorphic? The answer is that the dimensions tell us all.

Theorem 6.3.5
Let V and W be finite-dimensional vector spaces over a field
F. Then V and W are isomorphic if and only if dim(V) =
dim(W).

Proof
Suppose first that dim(V) = dim(VF) = n. If n = 0, then V
and W are both zero spaces and hence are surely isomorphic.
Let n > 0. Then V and W have bases, say {vi,..., v n } and
{wi,..., w n } respectively. There is a natural candidate for an
isomorphism from V to W, namely the linear transformation
T : V -»• W defined by

T(c 1v 1 H h c n v n ) = ciwi H h c n w n .
It is straightforward to check that T is a linear transformation.
Hence V and W are isomorphic.
Conversely, let V and W be isomorphic via an isomor-
phism T : V —> W. Suppose that {vi,..., v n } is a basis of V.
In the first place, notice that the vectors T(vi),..., T"(v n) are
linearly independent; for if ciT(vi) + - • -+c n T(v n ) = 0 ^ , then
-
T(ciVi + - • - + c n v n ) = 0w• This implies that ciVi + - • +c n v n
belongs to Ker(T) and so must be zero. This in turn implies
that c\ = • • • = c n = 0 because v i , . . . , v n are linearly inde-
pendent. It follows by 5.1.1 that dim(W) > n = dim(V). In
the same way it may be shown that dim(W) < dim(V^); hence
dim(V) = dim{W).
Corollary 6.3.6
Every n-dimensional vector space V over a field F is isomor-
n
phic with the vector space F .
For both V and F n have dimension n. This result makes
it possible for some purposes to work just with vector spaces
of column vectors.