Isomorphism is an equivalence relation on finite dimensional vector spaces over $F$.

Question

Show that isomorphism is an equivalence relation on finite dimensional vector spaces over $F$.

A relation $R$ is an equivalence relation if it is:

1. reflexive, i.e. $xRx$ for all $x$
2. symmetric, i.e. if $xRy$ then $yRx$
3. transitive, i.e. if $xRy$ and $yRz$ $\rightarrow$ $xRz$

If it's an isomorphism then:

$f(a)+f(b)=f(a+b)$ and $f(ab)=f(a)f(b)$

I'm a bit not clear what I need to do? Do I need to show that $aRa$ for $a\in F$ or $vRv$ for all $v \in V$? If so, where would isomorphic properties come in?

Answer 1

You need to show that given three $F$-vector spaces $X, Y, Z$:

• $X \cong Y$ implies $Y \cong X$;
• $X \cong Y, Y \cong Z$ implies $X \cong Z$;
• $X \cong X$.

The equivalence relation is between spaces, not between points in the spaces.