How to prove $\Phi(T)$ is an isomorphism

Question

I was trying to prove the theorem about $\Phi$ on my textbook (Linear Algebra $4^\text{th}$ edition : Theorem $2.20$ p.$103$).

The theorem:

Let $V$ and $W$ be finite-dimensional vector spaces over $F$ of dimensions $n$ and $m$, respectively, and let $\beta, \gamma$ be ordered bases for $V$ and $W$ respectively. Then the function $\Phi:L(V,W) \to M_{m \times n}(F)$,defined by $\Phi(T)=[T]^{\gamma}_{\beta}$ for $T \in L(V,W)$, is an isomorphism.

There is a proof on my book but I'm having trouble understanding it.

So it first mentioned that $\Phi$ is linear, which can be proved (Not sure how to prove it).

Then it says that we need to show $\Phi$ is one-to-one and onto (why both? if we can prove $\Phi$ is linear then one-to-one is equivalent to onto, right?)

Finally it says, it is accomplished if we can show: for every $m \times n$ matrix $A$ there is a unique linear transformation $T:V \to W$. Then conclude that $\Phi$ is an isomorphism?

Could anyone help with above questions? Many Thanks!

Answer 1

I'm assuming the book you're using is Friedberg, Insel, and Spence.

First, $\Phi$ is linear by Theorem 2.8 on page 82: $$\Phi(a\mathsf{T} + \mathsf{U}) = [a\mathsf{T}+\mathsf{U}]_\beta^\gamma = [a\mathsf{T}]_\beta^\gamma + [\mathsf{U}]_\beta^\gamma = a[\mathsf{T}]_\beta^\gamma + [\mathsf{U}]_\beta^\gamma = a\Phi(\mathsf{T}) + \Phi(\mathsf{U}).$$ Theorem 2.8 is what allows you to write the second and third equals signs above. The rest of the equals signs come from the definition of $\Phi$.

Your next two questions are addressed by reading the two sentences together. Here's the full quote:

Hence we must show that $\Phi$ is one-to-one and onto. This is accomplished if we show that for every $m \times n$ matrix $A$, there exists a unique linear transformation $\mathsf{T}:\mathsf{V} \to \mathsf{W}$ such that $\Phi(\mathsf{T}) = A$.

When you have a linear map between spaces of the same finite dimension, to show it's an isomorphism, it's enough to prove one of "one-to-one" or "onto". However, at this point in the book, you do not yet know that $\mathcal{L}(\mathsf{V}, \mathsf{W})$ is finite-dimensional when $\mathsf{V}$ and $\mathsf{W}$ are. The goal of Theorem 2.20 is to show that $\mathcal{L}(\mathsf{V}, \mathsf{W})$ is isomorphic to $\mathsf{M}_{m\times n}(F)$ by constructing the isomorphism $\Phi$. Thus we have to show both "one-to-one" and "onto": this is accomplished just as the quote says.