I don't understand the proof of the following problem:
Problem: Prove that the set S of all bilinear forms on a vector space V (call them Bil(V )) has the structure of a vector space.
Here's the proof:
"The natural addition and scalar multiplication on bilinear forms correspond with the addition and scalar multiplication of n × n matrices. We know that forms a vector space."
My question 1: Here the "natural addition and scalar multiplication" were defined elsewhere and I could understand them, what I don't understand is why the fact that these operations correspond to analogous operations of n × n matrices proves S is also a vector space.
My question 2: Also, does that mean if a set V and a vector space W are isomorphic (let $\Phi$ be an isomorphism between them) and V has the operations "addition $\star$" and "scalar multiplication $\lozenge$" such that $\Phi (a\star b)=\Phi(a)+\Phi(b)$ and $\Phi (\lambda \lozenge a)=\lambda \cdot \Phi(a)$, where + and $\cdot$ are addition and scalar multiplication on W, then V is also a vector space?
It's a bit misguided to talk about a set being a vector space or not. A vector space, strictly speaking, is a set together with the two vector space operations.
This means that to ask if a set is a vector space, you first have to define which two operations you want to use as the vector space operations, otherwise the question is meaningless.
It is similarly meaningless to speak of a set and a vector space being isomorphic, but if you have a bijection $\Phi$ from a set $S$ to a vector space $V$ and define operations $\star, \lozenge$ by $$ a \star b := \Phi^{-1}(\Phi(a) + \Phi(b)), \\ \lambda \lozenge a := \Phi^{-1}(\lambda \cdot \Phi(a)), $$ then your set $S$ with these operations will be a vector space. Basically, what that's doing is just "relabling" the elements of $V$ using the set $S$, which works since there is a one-to-one correspondence from elements of $V$ to elements of $S$ (given by $\Phi$).