Uniqueness of isomorphism to linear spaces.

Question

Even if an isomorphism between two linear spaces $L$ and $M$ over a field $\mathbb{K}$ exists, it is defined uniquely only in two cases:

1. $L=M=\{0\}$ and

2. $L$ and $M$ are one-dimensional, while $\mathbb{K}$ is a field consisting of two elements.

How can I show this fact? Does anyone have any hints?

Answer 1

For $2$-dimensions and above there are more than one isomorphism because you can map basis elements to basis elements and there are more than one way to do that.

For $1$-dimensional spaces, there is a map from the non-zero elements of the field to the set of automorphisms of any one-dimensional spaces induced by multiplication. Precomposing any isomorphism with an automorphism will change the isomorphism. So, if the field has more than two elements you can get different isomorphisms.

Answer 2

Hints

1. is extreamly easy. Recall that if $f,g:L\to M$ then $f=g$ if and only if $\forall v\in L,\,f(v)=g(v)$, so you can prove it only with set theory (an isomorphism is a bijection).

2. I guess you know that if $f\in\mathcal{L}(L,M)$ and $L$ is finite dimensional then $f$ is totally defined only by knowing the images of a basis of $L$, right? So you can use this.