TITLE: Uniqueness of isomorphism to linear spaces.
QUESTION [1 upvotes]: 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:

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

$L$ 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?

REPLY [1 votes]: Hints

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).
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.
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