Show that $\mathbb F^{ \;m \times n}$ is isomorphic to $\mathbb F^{m n}$,where $\mathbb F$ is a field.

Question

Show that $\mathbb F^{ \;m \times n}$ is isomorphic to $\mathbb F^{m n}$,where $\mathbb F$ is a field.

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We can show that both have the same dimension: $$\text{dim}(\mathbb F^{ \;m \times n})=mn=\text{dim}(\mathbb F^{m n})$$

On the other hand from this link we know there exists an isomorphism between the two vector spaces,hence the result.

I'm not sure if my argument is true.

Answer 1

Your argument is correct.

If you denote $E_{ij}$ the matrix having all entries equal to zero except the one at $i$th-row and $j$th-column which is equal to one, then $$\{E_{ij} \mid 1 \le i \le m, \, 1 \le j \le n\}$$ is a basis of $\mathbb F^{m \times n}$. Therefore $\dim \mathbb F^{m \times n} = mn.$