\paragraph{Exercise 5.1.} Suppose $B = \{v_1, \ldots, v_n\}$ is an orthogonal basis of $\mathbb{R}^n$. Show that $B$ is a basis of $\mathbb{R}^n$. 

\paragraph{Exercise 5.2.} Let $A \in \mathbb{R}^{n \times n}$ be an orthogonal matrix. Show that
\begin{enumerate}
\item[(a)] for any $x \in \mathbb{R}^n$: $\|Ax\| = \|x\|$,
\item[(b)] for any $x,y \in \mathbb{R}^n$: $\langle Ax, Ay \rangle = \langle x, y \rangle$.
\end{enumerate}

\paragraph{Exercise 5.3.} Let $s_\tau : \mathbb{R}^n \to \mathbb{R}^n$ be the function that cyclically shifts the entries of $x$ by $\tau$, $s_\tau(x) := (x_\tau, x_{\tau+1}, \ldots, x_n, x_1, \ldots, x_{\tau-1})^T$.
\begin{enumerate}
\item[(a)] Construct its matrix $S_\tau \in \mathbb{R}^{n \times n}$.
\item[(b)] Which properties does $S_\tau$ have? Mark all options that are correct (and only those):
\begin{itemize}
\item $S$ is square,
\item $S$ is symmetric,
\item $S$ is invertible,
\item $S$ is orthogonal,
\item $S$ is a multiple of the identity matrix,
\item $S$ could be the adjacency matrix of a graph.
\end{itemize}
\end{enumerate}

\paragraph{Exercise 5.4.} Let $v^{(1)}, \ldots, v^{(n)}$ be an orthonormal basis of $\mathbb{R}^n$. For any $x \in \mathbb{R}^n$, let $y \in \mathbb{R}^n$ be the vector of coefficients of $x$ represented with respect to $v^{(1)}, \ldots, v^{(n)}$ (i.e. $y = (y_1, \ldots, y_n)^T$ such that $x = \sum_{j=1}^n y_j v^{(j)}$).
\begin{enumerate}
\item[(a)] Show: $\|y\| = \|x\|$.
\item[(b)] Let $\tilde{y} := \sum_{j=1}^k y_j v^{(j)}$ for $k < n$, and assume that $\sum_{j=1}^k (y_j)^2 \ge 0.9 \sum_{j=1}^n (y_j)^2$. What can you say about $\|\tilde{y}\|$ in comparison to $\|y\|$?
\end{enumerate}

\paragraph{Exercise 5.5.} Let $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathbb{R}^{2 \times 2}$. Show that $\lambda \in \mathbb{R}$ is an eigenvalue of $A$ if and only if $\lambda$ satisfies the equation
\[
\lambda^2 - (a + d)\lambda + ad - bc = 0.
\]
Give a formula to compute the solutions of this equation.

\paragraph{Exercise 5.6.} Let $A \in \mathbb{R}^{m \times n}$ be a square matrix.
\begin{enumerate}
\item[(a)] Suppose there is a matrix $B \in \mathbb{R}^{n \times m}$ (note that the numbers of rows and of columns, respectively, are swapped!) such that $BA = I \in \mathbb{R}^{n \times n}$ is the $n \times n$ identity matrix; such a matrix $B$ is called a left inverse of $A$. Show that for every $b \in \mathbb{R}^m$, the system of linear equations
\[
Ax = b
\]
has a solution. Is the solution necessarily unique, or can there be more than one?

\item[(b)] Let $A \in \mathbb{R}^{n \times n}$. Show that $A$ is invertible if and only if for every $b \in \mathbb{R}^n$, the system of linear equations $Ax = b$ has exactly one solution $x \in \mathbb{R}^n$.

\item[(c)] Let $A \in \mathbb{R}^{m \times n}$, and suppose that by applying elementary row operations to $A$, we obtain a matrix $A'$ in row echelon form. Consider the following statements:
\begin{enumerate}
\item[(i)] For every $b \in \mathbb{R}^m$, the system of equations $Ax = b$ has a solution $x \in \mathbb{R}^n$;
\item[(ii)] The matrix $A'$ has rank $m$, i.e., it has precisely $m$ pivot elements.
\end{enumerate}
Show that these are equivalent, i.e., either one of them implies the other one. Show also that (ii) implies that $m \le n$.

\item[(d)] Let $A$ and $A'$ be as in Part (c). Consider the following statements:
\begin{enumerate}
\item[(iii)] For every $b \in \mathbb{R}^m$, the system of equations $Ax = b$ has exactly one solution $x \in \mathbb{R}^n$;
\item[(iv)] $m = n$, and the matrix $A'$ has rank $n$, with $n$ nonzero entries on the diagonal and zero entries below the diagonal, and arbitrary entries above the diagonal (i.e., the entries $a'_{ij}$ of $A'$ satisfy $a'_{ii} \ne 0$ and $a'_{ij} = 0$ for $i > j$, where $1 \le i,j \le n$).
\end{enumerate}
Show that the statements (iii) and (iv) are equivalent.
\end{enumerate}