Worksheet 14

1. If \(A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 3 & 2 \\ 3 & 2 & 1 \end{pmatrix}\) then \(\langle v, w \rangle := v^T A w\) is an inner product on $\mathbb{R}^3$?

2. Let $\langle -,- \rangle_1,\langle -,- \rangle_2$ be inner products on two vector spaces $V, W$. Show that \(\langle (v_1,w_1), (v_2,w_2) \rangle := \langle v_1,v_2 \rangle_1 + \langle w_1, w_2 \rangle_2\) is an inner product on $V \times W$.

3. Let $V$ be a vector space. Consider the set \(\operatorname{Pairings}(V) := \lbrace \langle -,- \rangle : V \times V \to k \mid \langle -,- \rangle \text{ is a bilinear and symmetric } \rbrace.\) Does addition \(\langle v,w \rangle_1 + \langle v,w \rangle_2\) and scalar multiplication \(c\langle v,w \rangle\) make $\operatorname{Pairings}(V)$ into a vector space? Is the subset of inner products in $\operatorname{Pairings}(V)$ a subspace?