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\newcommand{\bu}{{\mathbf{u}}}
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% other useful stuff
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\begin{document}
\begin{center}
\Large{Exercises for Module 8: Linear Algebra II}
\end{center}


1. Let $D \in \mathcal{L}(\mathbb{P}_4(\R),\mathbb{P}_3(\R))$ be the differentiation map, $Dp = p'$. Find bases of $\mathbb{P}_4(\R)$ and $\mathbb{P}_3(\R)$ such that the matrix representation of $\mathcal{M}(D)$ with respect to these basis is given by 
    \begin{equation*}
    \mathcal{M}(D) = \begin{pmatrix}
    1 & 0 & 0 & 0 & 0 \\
    0 & 1 & 0 & 0 & 0 \\
    0 & 0 & 1 & 0 & 0 \\
    0 & 0 & 0 & 1 & 0
     \end{pmatrix}.
    \end{equation*}

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2. Show that matrix multiplication of square matrices is not commutative, i.e find matrices $A,B \in M_{2}$ such that $AB\neq BA$.
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3.  A square matrix is called \emph{nilpotent} if $\exists k \in \N$ such that $A^k = 0$. Show that for a nilpotent matrix $A$, $|A| = 0$.

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4. A real square matrix $Q$ is called \emph{orthogonal} if $Q^T Q = I$. Prove that if $Q$ is orthogonal, then $|Q| = \pm 1$.

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5. An $n \times n$ matrix is called \emph{antisymmetric} if $A^T = -A$. Prove that if $A$ is antisymmetric and $n$ is odd, then $|A|=0$. 

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6. Let $V$ be an inner product space, $U$ a vector space and $S\colon U \to V$, $T\colon U \to V$ be linear maps. Show that $\innerprod{S\bu,\bv}= \innerprod{T\bu, \bv}$ for all $\bu \in U$ and $\bv \in V$ implies $S=T$.

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%7. Let $U,V,W$ be inner product spaces and $S,T \in \mathcal{L}(U,V)$ and $R\in \mathcal{L}(V,W)$. Show that the following holds
%\begin{enumerate}
%    \item $(S+\alpha T)^* = S^* + \overline{\alpha}T^*$ for all $\alpha\in \F$
%    \item $(S^*)^* = S$
%    \item $(RS)^* = S^*R^*$
%    \item $I^* = I$, where $I \colon U \to U$ is the identity operator on $U$
%\end{enumerate}
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