Due date: Monday 09. December 2024, 10 AM.
Exercise 1
Let \(V\) be a finite dimensional \(\mathbb{K}\)-vector space and \(f : V \to V\) an endomorphism. An ordered basis \(\mathbf{b}\) of \(V\) is called a Jordan basis for \(f\) if \[\mathbf{M}(f,\mathbf{b},\mathbf{b})=\operatorname{diag}(\mathbf{J}_{n_1}(\lambda_1),\ldots,\mathbf{J}_{n_k}(\lambda_k))\] where \(n_1,\ldots,n_k\) are integers and \(\lambda_1,\ldots,\lambda_k \in \mathbb{K}\) and where we write \[\mathbf{J}_{n}(\lambda)=\begin{pmatrix} \lambda & 1 & & & \\ & \lambda & 1 & & \\ & & \ddots & 1 & \\ & & & \lambda & 1 \\ & & & & \lambda\end{pmatrix} \in M_{n,n}(\mathbb{K}).\] Suppose \(\mathbf{b}=(v_1,\ldots,v_n)\) is a Jordan basis for \(f.\)
Describe \(\mathbf{M}(f\circ f,\mathbf{b},\mathbf{b}).\)
Let \(\mathbf{c}=(v_n,v_{n-1},\ldots,v_2,v_1).\) Describe \(\mathbf{M}(f,\mathbf{c},\mathbf{c}).\)
Exercise 2
Let \(U\subset V\) be subspaces of the \(n\)-dimensional \(\mathbb{K}\)-vector space \(W.\)
Let \(\Xi : W\to (W^*)^*\) be the isomorphism (see Exercise 13.9 in the lecture notes) given by the rule \(\omega \hspace{3pt}\rule[0.5pt]{2mm}{0.5pt}\rule[0.5pt]{0.5pt}{4.5pt}\hspace{3pt}\Xi(w) = w\hspace{3pt}\rule[0.5pt]{2mm}{0.5pt}\rule[0.5pt]{0.5pt}{4.5pt}\hspace{3pt}\omega = \omega(w).\) Show that \[(U^0)^0=\Xi(U).\]
Show that \(V^0\subset U^0.\)
Exercise 3
Let \(V=M_{n,n}(\mathbb{R})\) and let \(\mathbf{A}\in V.\) Define \(T_\mathbf{A}:V\to\mathbb{R}, \mathbf{B}\mapsto\operatorname{Tr}(\mathbf{A}^T\mathbf{B}).\)
Show that \(T_\mathbf{A}\in V^*.\)
Show that every element \(\nu\in V^*\) arises as above, i.e. \(\nu = T_\mathbf{A}\) for some \(\mathbf{A}\in V.\)
Show that the map \(\mathbf{A}\mapsto T_\mathbf{A}\) is an isomorphism from \(V\) to \(V^*.\)
Exercise 4
Let \(p : \mathbb{K}\to \mathbb{K}\) be a polynomial defined by the rule \(p(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0\) for all \(x \in \mathbb{K}.\) For a matrix \(\mathbf{A}\in M_{n,n}(\mathbb{K})\) we write \[p(\mathbf{A})=a_n\mathbf{A}^n+a_{n-1}\mathbf{A}^{n-1}+\cdots +a_1\mathbf{A}+a_0\mathbf{1}_n.\] Find for all \(\mathbf{A}\in M_{2,2}(\mathbb{K})\) a quadratic polynomial \(q : \mathbb{K}\to \mathbb{K}\) so that \(q(\mathbf{A})=\mathbf{0}_2.\) Does \(q\) look familiar?