2.3. Definition If $(V,b)$ is a bilinear space, then $$ \hat{b} \colon V \to V^* $$ defined by $$ (\hat{b} x)(y) = b(x,y) $$ is a obviously a linear transformation. $\hat{b}$ is called the adjoint transformation.
I want to show that $\hat{b}$ is linear map so take element $x, z$, so $$ \hat{b}(x+z)y = b(x+z,y) = b(x,y) + b(y,z) $$ and similarly $$ \hat{b}(cx)y = b(cx,y) = cb(x,y) \,. $$ Is it correct or not? Please help regarding this. I also want to know one example regarding adjoint transformation of bilinear form.
Of course it is linear.
The standard example is the bilinear form on $\Bbb R^n$: $$b(x, y) =\langle x, y\rangle=x_1y_1+\dots+x_ny_n$$ In particular, for the $i$th basis vector $e_i$, we get $\hat be_i=y\mapsto y_i\ (\Bbb R^n\to\Bbb R) $ is the $i$th coordinate map.
And for $x=(x_1,\dots, x_n)^T=x_1e_1+\dots+x_ne_n$ we get $$\hat bx=x_1\hat be_1+\dots+x_n\hat be_n$$ Note also that for a finite dimensional vector space $V$, if $b$ is a nondegenerate synmetric bilinear form (meaning $\forall y:b(x, y) =0\, \implies x=0$), then $\hat b$ is an isomorphism between $V$ and $V^*=\{f:V\to\Bbb R\mid f\ \text{linear} \} $.