About the proof of $\mathcal{L}(V,W)$ is isomorphic to $M_{n\times m}(F),$ given $\dim{V}=n,\dim{W}=m$?

Question

In the proof it just shows that $\large\Phi$ is onto, so why the part of one-to-one(injective) is omitted? And I'm taking an online course the teacher said: If $T:V\to W$ linear, and $V,W$ finite dimension then onto implies one-to-one. But as the second pic. theorem2.5 it says that it should be finite and of equal dimension, so which part of my understanding is wrong? Or finite dimension implies equal dimension in theorem 2.5?

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Answer 1

The author says "... there exists a unique linear transformation $T: V \to W$..." in the proof. "There exists" is surjectivity, "unique" is injectivity. The quoted theorem by the author (Theorem 2.6 on page 72) gets the uniqueness for you.