Page 653 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Figure 8.6.1

If , then we say that V and have the same algebraic structure. This means that although the names conventionally
given to the vectors and corresponding operations in V may differ from the corresponding traditional names in , as vector
spaces they really are the same.

Isomorphisms between Vector Spaces

It is easy to show that compositions of bijective linear transformations are themselves bijective linear transformations. (See the
exercises.) This leads to the following theorem.

THEOREM 8.6.3

Isomorphism of Finite-Dimensional Vector Spaces , then V and W are isomorphic.
Let V and W be finite-dimensional vector spaces. If

Proof We must show that there is an isomorphism from V to W. Let n be the common dimension of V and W. Then there is an

isomorphism by Theorem 8.6.2. Similarly, there is an isomorphism . Let . Then is an

isomorphism from V to W, so V and W are isomorphic.

EXAMPLE 4 An Isomorphism between and

Because and , these spaces are isomorphic. We can find an isomorphism T between them by

identifying the natural bases for these spaces under :

If is in , then by linearity,