Linear transformation representation proof

Question

I am wanting for someone to go over what I have and possibly correct my mistakes. Or any comments on the techniques, etc.

I want to prove that if $V$ and $W$ are vector spaces over some field $F$, with dimensions $n$ and $m$ respectively, bases $\alpha$, $\beta$ respectively then

$G: L(V,W) \to M_{m \times n}(F)$ defined by $G(z)=[z]^{\alpha}_{\beta}$ is an isomorphism.

What I tried;

To show that $G$ is linear must show that

for $f_{1}, f_{2}$ in $L(V,W)$

$$G(\lambda_{1}f_{1}+\lambda_{2}f_{2})=\lambda_{1}G(f_{1})+\lambda_{2}G(f_{2})$$

Which I attempted to show by essentially just writing out the definitions of matrix representation and then using that $f_{1},f_{2}$ are linear etc , expanding the matrix and factoring out the constants. Is that all I would need for that?

Now to show injectivity and surjectivity;

Injectivity can be shown if $\ker(G)=\{0\}$

If $g \in \ker(G)$ then its matrix representation is the zero matrix, which implies that $g$ must be zero as the matrix representation consists of those scalars used to write $g$, and those scalars are from a basis, i.e. linearly independent.

For surjectivity, I think I would need to show that for any $A \in M(F)$, there exist some $g$ in $L(V,W)$ with $A$ as its representation.

How does this seem so far? Is it correct/incorrect? Comments/advice?

Should I start over ? All I am looking for is any advice . Please guys ,Thank you

Answer 1

I'm not really following your descriptions at key points, particularly here:

Which I attempted to show by essentially just writing out the definitions of matrix representation and then using that $f1,f2$ are linear etc , expanding the matrix and factoring out the constants. Is that all I would need for that?

This sounds like right, but is so vague a description, it could be hiding a host of errors. If your proof starts off with $G(\lambda_1f_1 + \lambda_2f_2)$ and shows step-by-step that this is equal to $\lambda_1G(f_1) + \lambda_2G(f_2)$, or if it goes the other way, then you are good. However, if you work on both sides and produce the same thing out of them (something often seen out of students at a certain level, even in college), then at least, you've got some rewriting to do. Possibly some refiguring.

If g∈Ker(G) then its matrix representation is the zero matrix, which implies that g must be zero as the matrix representation consists of those scalars used to write g, and those scalars are from a basis ie linearly independent.

This sounds a lot like circular reasoning - you are using the result you are trying to prove to prove it. But I can't say this for sure, as the wording is again too vague to understand your approach. Another possibility is that you are waving your hands too much and skipping over the things you need to show. I just can't tell.

For surjectivity, I think I would need to show that for any A∈M(F), there exist some g in L(V,W) with A as its representation.

Yes. That is the definition of surjectivity.

To show it, for $v \in V$, express it in terms of the basis $\alpha:\ v = \sum_i v_i\alpha_i$. Define $w_j$ by $[w_j] = A[v_i]$, and define $g(v) = \sum_j w_j\beta_j$. Show that $g$ is linear and $G(g) = A$.