Page 69 - Linear Algebra Done Right

Invertibility
T(aSw) = aT(Sw) = aw.
Thus aSw is the unique element of V that T maps to aw. By the
definition of S, this implies that S(aw) = aSw. Hence S is linear, as 55
desired.
Two vector spaces are called isomorphic if there is an invertible The Greek word isos
linear map from one vector space onto the other one. As abstract vector means equal; the Greek
spaces, two isomorphic spaces have the same properties. From this word morph means
viewpoint, you can think of an invertible linear map as a relabeling of shape. Thus
the elements of a vector space. isomorphic literally
If two vector spaces are isomorphic and one of them is finite dimen- means equal shape.
sional, then so is the other one. To see this, suppose that V and W
are isomorphic and that T ∈L(V, W) is an invertible linear map. If V
is finite dimensional, then so is W (by 3.4). The same reasoning, with
T replaced with T −1 ∈L(W, V), shows that if W is finite dimensional,
then so is V. Actually much more is true, as the following theorem
shows.

3.18 Theorem: Two finite-dimensional vector spaces are isomorphic
if and only if they have the same dimension.

Proof: First suppose V and W are isomorphic finite-dimensional
vector spaces. Thus there exists an invertible linear map T from V
onto W. Because T is invertible, we have null T ={0} and range T = W.
Thus dim null T = 0 and dim range T = dim W. The formula

dim V = dim null T + dim range T

(see 3.4) thus becomes the equation dim V = dim W, completing the
proof in one direction.
To prove the other direction, suppose V and W are finite-dimen-
sional vector spaces with the same dimension. Let (v 1 ,...,v n ) be a
basis of V and (w 1 ,...,w n ) be a basis of W. Let T be the linear map
from V to W defined by
T(a 1 v 1 +· · ·+ a n v n ) = a 1 w 1 +· · ·+ a n w n .

Then T is surjective because (w 1 ,...,w n ) spans W, and T is injective
because (w 1 ,...,w n ) is linearly independent. Because T is injective and