My question comes from a proof of establishing that the function $\Phi: \mathcal{L}(V,W) \to M_{m\times n}(F)$ is an isomorphism. The statement of the theorem comes from Linear Algebra by Friedberg, Insel, and Spence. What follows is the statement of theorem and its proof.
I understand the uniqueness portion of this proof. What I'm not grasping is how showing the existence of this matrix establishes that $\Phi$ is one-to-one and onto?
Best Answer
For injective: assume $\Phi(T)=0$ then $[T]^\gamma_\beta=0 \Rightarrow T=0$. So $N(\Phi)=\{0\}$ $\Rightarrow$ $\Phi$ is injective.
For surjective: Take a matrix, define a linear transformation then how are these two the linear transformation and matrix are defined? The linear transformation must be such that the matrix of the linear transformation relative to the basis $\beta$ and $\gamma$ is the matrix that you started with.
Can you show that $\Phi$ is linear transformation?