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\begin{document}
\title{Math 241 Homework}
\author{Mary Radcliffe}
\date{Due 6 December 2018}
\maketitle

Complete the following problems. Fully justify each response.


\begin{enumerate}

\item Complete problems 12.2 and 12.3 on page 239 in \color{blue}\href{http://web.stanford.edu/~boyd/vmls/vmls.pdf}{Introduction to Applied Linear Algebra} (ALA) \color{black} by Stephen Boyd and Lieven Vandenberghe. (Note: when we think about angles between vectors, you may use the property, not proved in this class, that if $\theta$ is the angle between $\vec x$ and $\vec y$, then $\vec x^T\vec y = \|\vec x\|\|\vec y\|\cos\theta$.)
\item Complete problems 13.7, 13.8, 13.10 on pages 280-281 in \color{blue}\href{http://web.stanford.edu/~boyd/vmls/vmls.pdf}{Introduction to Applied Linear Algebra} (ALA) \color{black} by Stephen Boyd and Lieven Vandenberghe.

\item Let $A$ be an $m\times n$ matrix with singular value decomposition $A=U\Sigma V^T$, where $U$ is $m\times m$, $\Sigma$ is $m\times n$, and $V$ is $n\times n$. Write the columns of $U$ as $\vec u_1, \vec u_2, \dots, \vec u_m$, the columns of $V$ as $\vec v_1, \vec v_2, \dots, \vec v_n$, and let $k=\rank A$. Let $\vec b\in \R^m$, and write $\vec b = \alpha_1\vec u_1+\dots+\alpha_m\vec u_m$ as a linear combination of the columns of $U$ (which are a basis for $\R^m$.

Prove that the least squares solution $\hat{\vec x}$ to the equation $A\vec x = \vec b$ satisfies $A\hat{\vec x} = \alpha_1\vec u_1+\dots+\alpha_k\vec u_k$; that is, it is the terms in $\vec b$ corresponding to the nonzero singular values of $A$. Show that $\hat{\vec x}= \frac{\alpha_1}{\sigma_1}\vec v_1 + \dots + \frac{\alpha_k}{\sigma_k}\vec v_k$ satisfies this equation.

(Notice that in the above analysis, we did NOT rely on independence or dependence of the columns of matrix $A$. So we can use the SVD for $A$ to give solutions to a least squares problem regardless of these properties).

\item Complete the problem set found at \url{autolab.andrew.cmu.edu}. The submission for this is directly on autolab, no need to hand it in on paper.

\end{enumerate}
\end{document}