Show that two finite dimensional vector spaces over F are isomorphic iff. they have equal dimensions.
Suppose X, Y are isomorphic where X, Y are finite dimensional vector spaces.
Since f is isomorphic, f is monic and an epimorphism. Therefore, f is in a one to one correspondence. Thus, dim X = dim Y.
Suppose dim X = dim Y.
Define by .
If , then
Therefore, f is monic.
There exist such that . Therefore, f is an epimorphism.
Hence, f is an isomorphism.