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December 6th, 2016, 06:02 PM   #1
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Bilinear Transformations

**Question**

Let V and W be vector spaces over a field F and let $T \in Hom (V,W)$. For each $g \in Bil(W\times W)$, define $g(T): V \times V \to F$ by setting $g(T): (x, y) \mapsto (T(x), T(y))$.

a) Prove that $g(T) \in Bil (V \times V)$.

b) Prove that $K:Bil(W \times W) \to Bil (V \times V)$ defined by setting $g \mapsto g(T)$ is a linear transformation.

**My Attempt**

a) In order to prove bilinear we need to show that $$f(au_1+b⋅u_2,v)=af(u_1,v)+bf(u_2,v)$$
$$f(u,av_1+bv_2)=af(u,v_1)+bf(u,v_2)$$

Let $x=\langle x_1,x_2\rangle$ and $y=\langle y_1,y_2\rangle$, then

$\begin{align}g(ax_1+bx_2,y)&=g(a\langle x_{11},x_{12}\rangle+b\langle x_{21},x_{22}\rangle,\langle y_1,y_2\rangle)\\
&=g(T(a\langle x_{11},x_{12}\rangle+b\langle x_{21},x_{22}\rangle),T(\langle y_1,y_2\rangle))\\
&=g(T(a\langle x_{11},x_{12}\rangle)+T(b\langle x_{21},x_{22}\rangle)),T(\langle y_1,y_2\rangle))\end{align}$

This is where I get stuck. I've never done this with transformations, can I just split this and distribute (for lack of a better term) the T?

Also, for b, how does the bilinear affect the proof for a linear transformation?
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