Let $\langle \ ,\ \rangle : \mathbb{R}^2 \times \mathbb{R}^2 \rightarrow \mathbb{R} $ given by: $$ ⟨(u_1, u_2),(v_1, v_2)⟩ = u_1(v_1 + v_2) + u_2v_2 $$
$1$. Is it bilinear? And symmetric?
$ ⟨(a (u_1, u_2) + b (v_1 + v_2) , (w_1, w_2)⟩ = a ⟨u,w⟩ + b ⟨v,w⟩ $, $ \ \forall u,v, w \in \mathbb{R}^2$ and $ \forall a,b\in \mathbb{R} $
$ ⟨(u_1, u_2) , (a (v_1, v_2) + b (w_1 + w_2) ⟩ = a ⟨u,v⟩ + b ⟨u,w⟩ $, $ \ \forall u,v, w \in \mathbb{R}^2$ and $ \forall a,b\in \mathbb{R} $
So $\langle \ ,\ \rangle$ is bilinear, but
$ ⟨u,v⟩ \neq ⟨v,u⟩ $, $\ \forall u,v \in \mathbb{R}^2$
So $\langle \ ,\ \rangle$ is not symmetric.
$2$. Find bilineal forms $⟨ \ ,\ ⟩_+,⟨\ ,\ ⟩_−$ such that:
(a) $⟨ \ ,\ ⟩ = ⟨\ ,\ ⟩_+ + ⟨\ ,\ ⟩_−$
(b) $⟨u, v⟩_+ = ⟨v, u⟩_+$
(c) $⟨u, v⟩_− = −⟨v, u⟩_−$
$ \forall u,v \in \mathbb{R}^2$
Find the matrix for $⟨ \ ,\ ⟩$, $⟨ \ ,\ ⟩_+$ and $⟨\ ,\ ⟩_−$ in canonical basis.
Is it even possible to find $⟨\ ,\ ⟩_+$ and $⟨\ ,\ ⟩_−$ that fits in (a), (b) and (c) at the same time? Any help to find that bilinear forms? Thanks!