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\begin{document}
\begin{center}
\Large{Exercises for Module 9: Linear Algebra III}
\end{center}

1. 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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2. Let $V$ be an inner product space and $\bx_1,\ldots, \bx_n$ be an orthonormal basis and $\by \in V$. Then, $\bx$ has a unique representation $\by =\sum_{i=1}^n \alpha_i \bx_i$. Show that $\alpha_i = \innerprod{\by,\bx_i}$ for all $i= 1,\ldots, n$.
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3. Let $V$ be an inner product space and $U\subseteq V$ a subset. Show that $U^\perp$ is a subspace of $V$.
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4.  Let $A,B \in M_n(\F)$ be similar matrices. Show that their characteristic polynomials coincide.

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5. Show that $A\in M_n(\C)$ is invertible if and only if $0 \not \in \sigma(A)$.

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6. Suppose $N$ is a nilpotent matrix. Show that $\sigma(N) = \{0\}$.

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7. Let $A\in M_n(\C)$ be an invertible matrix. Show that $\lambda$ is an eigenvalue of $A$ if and only if $\lambda^\inv$ is an eigenvalue of $A^\inv$.
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8. Suppose $A\in M_n(\C)$ is Hermitian. Show that all the eigenvalues of $A$ are real. Hint: Note that if $\bx$ is a normalized eigenvector of $A$ with eigenvalue $\lambda$, then $\innerprod{A\bx,\bx} = \lambda$.

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9. Let $A\in M_n(\R)$. Show that the eigenvalues of $A^TA$ are non-negative.
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