Let \(V\) be an \(\mathbb{R}\)-vector space. Show that every bilinear form \(\langle \cdot{,}\cdot\rangle : V\times V\to \mathbb{R}\) can be uniquely written as a sum of a symmetric and an alternating bilinear form.

Let \(n \in \mathbb{N}\) and \(\mathbf{A},\mathbf{B}\in M_{n,n}(\mathbb{R}).\) Show that if \(\vec{x}_1^T\mathbf{A}\vec{x}_2=\vec{x}_1^T\mathbf{B}\vec{x}_2\) for all vectors \(\vec{x}_1,\vec{x}_2 \in \mathbb{R}^n,\) then \(\mathbf{A}=\mathbf{B}.\)

Let \(\langle \cdot{,}\cdot\rangle\) be a symmetric bilinear form on a \(\mathbb{K}\)-vector space \(V,\) \(\mathbb{K}=\mathbb{R},\mathbb{C}.\) The bilinear form \(\langle \cdot{,}\cdot\rangle\) induces a so-called quadratic form \(q : V \to \mathbb{K},\) defined by the rule \(q(v)=\langle v,v\rangle\) for all \(v \in V.\) Show that we can reconstruct \(\langle \cdot{,}\cdot\rangle\) from \(q.\) That is, find an expression for \(\langle \cdot{,}\cdot\rangle\) which uses \(q\) only.

Let \(\vec v,\vec w\in\mathbb{R}^3\) and \(L_{\vec v}:\mathbb{R}^3\to\mathbb{R}^3,\) \(\vec u\mapsto \vec v\times \vec u.\) Show that \[\langle \cdot{,}\cdot\rangle: \mathbb{R}^3\times \mathbb{R}^3\to\mathbb{R}, \qquad (\vec v,\vec w)\mapsto \langle \vec v,\vec w\rangle = \mathrm{Tr}(L_{\vec v}\circ L_{\vec w})\] defines a symmetric bilinear form on \(\mathbb{R}^3\) by explicitly computing \(\mathrm{Tr}(L_{\vec v}\circ L_{\vec w}).\)