Due date: Monday 16. October 2023, 10 AM.
Exercise 1
We consider \[\mathbf{b}=\left(\begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix},\begin{pmatrix} 2 \\ 1 \\ 2 \end{pmatrix},\begin{pmatrix} 3 \\ 1 \\ 1 \end{pmatrix}\right)\qquad and \qquad \vec{v}=\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}.\]
Show that \(\mathbf{b}\) is an ordered basis of \(\mathbb{R}^3.\)
Compute the coordinates of the vector \(\vec{v}\) with respect to the linear coordinate system \(\boldsymbol{\beta}\) associated to \(\mathbf{b}.\)
Consider the ordered basis \[\mathbf{c}=\left(\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix},\begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix},\begin{pmatrix} 2 \\ 1 \\ 0 \end{pmatrix}\right).\] Compute \(\mathbf{C}(\mathbf{b},\mathbf{c}).\)
Compute the coordinates of the vector \(\vec{v}\) with respect to the linear coordinate system \(\boldsymbol{\gamma}\) associated to \(\mathbf{c}\) and verify that \[\boldsymbol{\gamma}(\vec{v})=\mathbf{C}(\mathbf{b},\mathbf{c})\boldsymbol{\beta}(\vec{v}).\]
Exercise 2
Find all linear maps \(f : \mathbb{R}^2 \to \mathbb{R}^2\) which map the line \(y=x\) surjectively onto the line \(y=3x.\)
Exercise 3
Consider the \(\mathbb{R}\)-vector space \(V\) of symmetric \(2\)-by-\(2\) matrices with real entries \[\mathbf{A}=\begin{pmatrix} a & b \\ b & c \end{pmatrix}\] and the mapping \[f : V \to M_{2,2}(\mathbb{R}), \qquad \mathbf{A}\mapsto \mathbf{B}\mathbf{A}\mathbf{B}^T,\] where \[\mathbf{B}=\begin{pmatrix} 2 & 1 \\ 0 & 1\end{pmatrix}.\]
Show that \(f\) is a linear map into \(V.\)
Compute the matrix representation of \(f\) with respect to a suitable basis of \(V.\)
Exercise 4
Let \(\mathbf{A}\in M_{m,n}(\mathbb{K}).\) Show that \(f_\mathbf{A}: \mathbb{K}^n \to \mathbb{K}^m\) and \(f_{\mathbf{A}^T} : \mathbb{K}^m \to \mathbb{K}^n\) have the same rank.