Does an injective $\mathbb F_9$ vector space homomorphism $\mathbb F_9^3 \to \mathbb F_9^5$ exist?

Question

Does an injective $\mathbb F_9$ vector space homomorphism $\mathbb F_9^3 \to \mathbb F_9^5$ exist?

Is it able to solve that task by some technique? If so, how is it working then?

I have posted a similiar question here but with a mapping that is not a homomorphism.

Answer 1

Yes. For example, consider the map which sends $(a,b,c)\in \mathbb{F}_9^3$ to $(a,b,c,0,0)\in \mathbb{F}_9^5$.

Answer 2

Let $\mathbb{F}$ be a field. If $\phi: \mathbb{F}^m \rightarrow \mathbb{F}^n$, and if $m \leq n$, then $\phi$ can be defined explicitly as an embedding, which is injective. (Simply annex $n - m$ zeros as components in your starting vector.) See here for more details.