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\fancyhead[L]{\textbf{MATH 510 Homework 6\qquad\qquad Name:}}
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{\small
\begin{itemize}
\item Homework 7 is due March 29 at the beginning of class.
\end{itemize}}

\begin{enumerate}[{\rm 1.}]
\item Let $\|\cdot\|$ be a norm (i.e. a ``vector norm'' as the book calls it) on $\C^n$. Let $T\in M_n(\C)$. Let $\|\cdot\|_T$ be the norm given by $\|x\|_T=\|Tx\|$. Show that if $T$ is an isometry with respect to $\|x\|$, then $\|\cdot\|^{\rm D}=(\|\cdot\|_T)^{\rm D}$, where $f(\cdot)^D$ denotes the dual norm of a (pre-)norm $f(\cdot)$.
\item Show that if $x,y\in\C^n$ then $\|y\|_{\infty}=\max_{\|x\|_1=1}|y^\ast x|$ and $\|y\|_1=\max_{\|x\|_\infty=1}|y^\ast x|$.
\item Prove that a norm $\|\cdot\|$ on a vector space $V$ over $\R$ (same is true for $\C$ but requires an extra step) is derived from an inner product $\langle\cdot,\cdot\rangle$ (in the sense that $\|x\|=\sqrt{\langle x,x\rangle}$ for all $x\in V$) if and only if $\|\cdot\|$ satisfies the \emph{parallelogram identity} 
\[\frac{1}{2}\big(\|x+y\|^2+\|x-y\|^2\big) = \|x\|^2+\|y\|^2,\qquad\forall x,y\in V.\]
\emph{Hint:} When $\|\cdot\|$ is derived from an inner product, $\langle x,y\rangle$ can be expressed in terms of $\|x+y\|$, $\|x\|$, and $\|y\|$. For the $\Leftarrow$ direction, use that formula as the definition of $\langle x,y\rangle$. To show additivity, use the parallelogram identity. For homogeneity $\langle \lambda x,y\rangle = \lambda \langle x,y\rangle$, first show it for $\lambda\in\Z$, then for $\lambda\in\Q$. Use Cauchy-Schwarz and a continuity argument for the general case $\lambda\in\R$.
\item Let $A\in M_n$ be a non-singular matrix such that the upper left $k\times k$-submatrix is singular (i.e. the leading principal $k\times k$-minor is zero) for some $k\in\{1,2,\ldots,n-1\}$. Show that $A$ cannot be factored as $LU$ where $L\in M_n$ is a lower-triangular matrix and $U\in M_n$ is an upper-triangular matrix. (\emph{Hint:} Start with small $k$ and arbitrary $n$ to see what happens.)
\item Against better judgement, call two matrices $A,B\in M_{m,n}$ \emph{equivalent} if there are non-singular matrices $S\in M_m$ and $T\in M_n$ such that $B=SAT$. 
\begin{enumerate}[{\rm (a)}]
\item Show that every matrix $A\in M_{m,n}$ is equivalent to a matrix of the form $\begin{bmatrix}I_k&0\\ 0&0\end{bmatrix}$ where $k\le\min\{m,n\}$ and $0$ are appropriately sized zero matrices. (\emph{Hint:} Use that row and column operations can be performed using matrix multiplication by elementary matrices and use induction. Alternatively, find appropriate bases for $\C^m, \C^n$.)
\item Show that two matrices in $M_{m,n}$ are equivalent if and only if they have the same rank.
\end{enumerate}
\end{enumerate}
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