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\begin{document}

\title{MATH 351, Fall 2017, Homework \#8}

\author{\underline{\bf Due Friday, November 3, 2017}}



\maketitle

\begin{problem} Write down an orthogonal basis for $\mathbb R^4$ using only the numbers $1$ and $-1$ as entries for the vectors.
\end{problem}

\begin{problem}[6.28, p.330] For any subset $S$ of $\mathbb R^n$ show that $$S^\perp =\{\vec x\in \mathbb R^n : \vec x\cdot \vec s = 0\ \textup{for all}\ \vec s\in S\}$$ is a subspace.
\end{problem}

\begin{problem} For any subspace $W$ of $\mathbb R^n$, show that $(W^\perp)^\perp = W$.
\end{problem}

\begin{problem}[cf. 6.22, p. 329] Use the Gram-Schmidt process to find an orthogonal basis for the subspace of $\mathbb R^n$ spanned by the following vectors.
\end{problem}

\begin{itemize}
\item[a)] $\{[1,2,1,2] ^t, [1,2,3,0]^t, [1,1,-1,-1]^t\}$
\item[b)] $\{[1,1,1,1,1]^t, [1,1,0,1,-1]^t, [1,-1,0,0,1]^t\}$
\end{itemize}

\begin{problem}[6.20, p. 330] Find an orthogonal basis for $S^\perp$ for each of these sets:
\end{problem}

\begin{itemize}
\item[a)] $S = \{[1,3,1,-1]^t, [2,6,0,1]^t, [4,12, 2,-1]^t\}$
\item[b)] $S = \{[1,3,1,-1]^t, [2,6,0,1]^t \}$
\item[c)] $S = \{[1,1,1,1,1]^t, [1,1,0,1,-1]^t, [1,-1,0,0,1]^t\}$
\end{itemize}

\begin{problem}[6.27, p. 330] Let $A$ be a $k\times n$ real matrix, and let $W$ be the null space of $A$. Show that $A\vec x = A(P_{W^\perp}(\vec x))$ for any $\vec x\in \mathbb R^n$, where $P_{W^\perp}(\vec x)$ is the orthogonal projection onto $W^\perp$.
\end{problem}








\begin{problem}[cf. 6.39, p. 331] Let $\vec x,\vec y\in \mathbb R^n$ be non zero vectors. Show that the orthogonal projection of $\vec x$ onto $\mathbb R\vec y$ is the vector of closest distance to $\vec x$ belonging to $\mathbb R\vec y$. Show this by:
\end{problem}

\begin{itemize}
\item[a)] minimizing the function $f(t) = \|\vec x - t\vec y\|^2$
\item[b)] using that for any subspace $W$ of $\mathbb R^n$, $\|\vec x + \vec y\|^2 = \|\vec x\|^2 + \|\vec y\|^2$ for $\vec x\in W$ and $\vec y \in W^\perp$.
\end{itemize}








\end{document}

\begin{problem}[cf. 6.32, p. 330] Let $W$ be a subspace of $\mathbb R^n$. If $\{\vec x_1, \dotsc, \vec x_k\}$ is an orthogonal basis for $W$ and $\{\vec y_1,\dotsc,\vec y_\ell\}$ is an orthogonal basis for $W^\perp$, then show that $\{\vec x_1,\dotsc,\vec x_k,\vec y_1,\dotsc,\vec y_\ell\}$ is an orthogonal basis for $\mathbb R^n$.
\end{problem}


For later...

\begin{problem} Let $P$ be an $n\times n$ real matrix such that $P^t = P$ and $P^2 = P$. Show that $P$ is the matrix of the orthogonal projection onto the subspace of all vectors $\vec x\in \mathbb R^n$ such that $P\vec x = \vec x$.
\end{problem}

