A question about the relation between linear transformations and their representing matrices

Question

I would like to have some clarification regarding the notion that all linear transformations can be represented by "representing matrices". There's this lemma that the space of all linear transformations from spaces V to U are isomophic to m×n matrices (m being dimV and n being dimU). Which leads to my question: given a matrix m×n, can I deduce that there exists some transformation from V to U (dimV = m and dimU =n) that is represented by this very matrix (since isomophic relationships go both ways)?

Answer 1

Yes that's right... though it seems your convention for matrix representations is different from what I'm used to. Lets say $V,U$ are vector spaces over a field $F$ of dimension $n,m$ respectively. Then, yes we can construct an isomorphism $L(V,U)\cong M_{m\times n}(F)$, but this isomorphism depends heavily on the choice of basis.

More explicitly, if $\beta$ is a basis of $V$, and $\gamma$ is a basis of $U$, then we can construct a linear isomorphism $\Phi_{\beta,\gamma}:L(V,U)\to M_{m\times n}(F)$. But, like I've indicated in the notation, the isomorphism $\Phi_{\beta,\gamma}$ very much depends on the basis $\beta,\gamma$. If you change the basis to a different one, say $\beta', \gamma'$, you will get a different isomorphism $\Phi_{\beta',\gamma'}$. So you always have to be careful when dealing with matrices and always make sure to explicitly mention what the bases being used are.