How to prove there is left\right adjoint for non-degenerate bilinear form?

Question

Let $V$ and $W$ be $\mathbb{F}$-vector spaces such that $dim(V) =dim(W)<∞$. Let $φ$ be a non-degenerate bilinear form on $V×W$.
Show that: for every $f∈L(V)$, there exists $g\in L(W)$ such that $φ(f(v),w) =φ(v,g(w))$, $v∈V,w∈W$. This is the right adjoint of $f$ and similarly, there is a left adjoint. I have no idea how to solve this.

Answer 1

Hint: Let $\{v_1,\dots,v_n\}$ and $\{w_1,\dots,w_n\}$ be bases for $V$ and $W$ (respectively). We want to define a linear map $g:W \to W$ by selecting vectors $g(w_j)$ so that for all $i$ and $j$, we have $$ \varphi(v_i,g(w_j)) = \varphi(f(v_i),w_j) $$ Having done this, we can extend $g$ to all of $W$ by linearity.

That is, for each $j = 1,\dots,n$, we wish to find vectors $g(w_j)$ which solve the linear system $$ \pmatrix{ \varphi(v_1,g(w_j))\\ \varphi(v_2,g(w_j))\\ \vdots\\ \varphi(v_n,g(w_j)) } = \pmatrix{ \varphi(f(v_1),w_j)\\ \varphi(f(v_2),w_j)\\ \vdots\\ \varphi(f(v_n),w_j) } $$

Using the fact that our bilinear form is non-degenerate, why is it necessarily possible to find such vectors $g(w_j)$?

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Hint 2: Show that because $\varphi$ is non-degenerate, the linear map $$ \Phi(x) = \pmatrix{ \varphi(v_1,x)\\ \varphi(v_2,x)\\ \vdots\\ \varphi(v_n,x) } $$ is invertible.