Exercise 19

These prove that isomorphism is an equivalence relation.

• Show that the identity map $\text{id}\colon V\to V$ is an isomorphism. Thus, any vector space is isomorphic to itself.

• Show that if $f\colon V\to W$ is an isomorphism then so is its inverse $f^{-1}\colon W\to V$. Thus, if $V$ is isomorphic to $W$ then also $W$ is isomorphic to $V$.

• Show that a composition of isomorphisms is an isomorphism: if $f\colon V\to W$ is an isomorphism and $g\colon W\to U$ is an isomorphism then so also is $g\mathbin{\circ}f\colon V\to U$. Thus, if $V$ is isomorphic to $W$ and $W$ is isomorphic to $U$, then also $V$ is isomorphic to $U$.