On \(V=M_{2,2}(\mathbb{R})\) consider the bilinear form \(\langle \cdot{,}\cdot\rangle\) defined by the rule \(\langle \mathbf{A},\mathbf{B}\rangle=\operatorname{Tr}(\mathbf{A}\mathbf{B})\) for all \(\mathbf{A},\mathbf{B}\in M_{2,2}(\mathbb{R}).\)

1. Compute \(\mathbf{M}(\langle \cdot{,}\cdot\rangle,\mathbf{b})\) where \(\mathbf{b}=(\mathbf{E}_{i,j})_{1\leqslant i,j\leqslant 2}.\)

2. Compute the signature of \(\langle \cdot{,}\cdot\rangle.\)

3. Find an orthogonal basis for \(\langle \cdot{,}\cdot\rangle.\)

4. Compute the signature of the restriction of \(\langle \cdot{,}\cdot\rangle\) to the subspace \(U\subset V\) consisting of matrices with trace \(0.\)

5. \(\heartsuit\) Compute the signature of the corresponding bilinear form on \(M_{n,n}(\mathbb{R}).\)

Let \[\vec{v}_1=\begin{pmatrix}1 \\ 0 \\ 0 \end{pmatrix},\quad \vec{v}_2=\begin{pmatrix}-2 \\ 1 \\ 0 \end{pmatrix},\quad \vec{v}_3=\begin{pmatrix}4 \\ -1 \\ -2 \end{pmatrix}\in\mathbb{R}^3.\] Find a matrix \(\mathbf{A}\in M_{3,3}(\mathbb{R})\) such that \(\{\vec{v}_1,\vec{v}_2,\vec{v}_3\}\) is an orthonormal basis of \(\mathbb{R}^3\) with respect to the symmetric bilinear form \(\langle \cdot{,}\cdot\rangle_\mathbf{A}.\)

Consider the symmetric bilinear form \(\langle \cdot{,}\cdot\rangle: \mathsf P_2(\mathbb{R})\times \mathsf P_2(\mathbb{R})\to\mathbb{R}\) given by \[\langle f,g\rangle \mapsto \int_{-1}^{1}x f(x)g(x)\,\mathrm dx.\]

1. Show that \(\langle \cdot{,}\cdot\rangle\) is degenerate.

2. Find an ordered basis \(\mathbf b\) of \(\mathsf P_2(\mathbb{R})\) such that \(\mathbf{M}(\langle \cdot{,}\cdot\rangle,\mathbf b)\) is diagonal.

Let \(U_1,U_2\) be subspaces of a vector space that is equipped with a symmetric bilinear form. Show that

1. \((U_1+U_2)^{\perp}=U_1^{\perp}\cap U_2^{\perp}\);

2. \(U_1\subset ((U_1)^{\perp})^{\perp}\);

3. If \(U_1\subset U_2,\) then we have \(U_2^{\perp}\subset U_1^{\perp}.\)