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

Complete the following problems. Fully justify each response.

This homework set can be turned in during lecture on 5 November, or may be turned in to Mary's office between 11:00 and 2:00 on Tuesday, 6 November. If these turn in times are not possible for you, please speak to your TA about an alternative.

\begin{enumerate}

\item Define a projection $P:\R^n\to\R^n$ to be a linear transformation such that $P^2=P$.
\begin{enumerate}
\item Prove that an orthogonal projection onto a subspace $W$ is a projection, as defined above.
\item Give an example of a projection that is NOT an orthogonal projection onto a subspace.
\item What are the possible eigenvalues of a projection? Explain.
\end{enumerate}

\item Complete problem 8.6.3 on page 350 of Coding the Matrix.

\item Prove that a set of mutually orthogonal, nonzero vectors is linearly independent.

\item Let $W$ be a subspace of $\R^n$. Define $W^\perp=\{\vec v\in \R^n\ | \ \vec{v}\perp W\}$. Prove that if $\mathcal{B}_W$ is a basis for $W$, and $\mathcal{B}_{W^{\perp}}$ is a basis for $W^\perp$, then $\mathcal{B}_W\cup\mathcal{B}_{W^\perp}$ is a basis for $\R^n$. (Hint: I would NOT use a dimension argument here. Instead, try an argument about unique representation of every vector in $\R^n$)

\item Let $\lambda_1, \lambda_2$ be two different eigenvalues for a symmetric matrix $A$, and let $\mathcal{E}(\lambda_1), \mathcal{E}(\lambda_2)$ be the corresponding eigenspaces. Prove that if $v\in \mathcal{E}(\lambda_2)$, then $v\perp \mathcal{E}(\lambda_1)$. Use this to show that if $A$ has a set of $n$ linearly independent eigenvectors, it also has a set of $n$ orthogonal eigenvectors.

\item (Optional turnin) Compute the following projections.
\begin{enumerate}
\item projection of $\vec{v}=\threevec{1}{3}{2}$ onto $\sp{\threevec{2}{4}{0}}$
\item projection of $\vec{v}=\threevec{1}{3}{-1}$ onto $\sp{\threevec{1}{2}{1},\threevec{1}{2}{-1}}$
\end{enumerate}

\item (Optional turnin) Find an orthogonal basis for each of the following.
\begin{enumerate}
\item The plane in $\R^3$ through the origin whose equation can be written as $2x+3y-z=0$.
\item The hyperplane (a 3-dimensional space) in $\R^4$ through the origin whose equation can be written as $x_1+2x_2-x_4=0$.
\end{enumerate}

\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}