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\begin{document}

\title{MATH 351, Fall 2017, Homework \#6}

\author{\underline{\bf Due Wednesday, October 18, 2017}}



\maketitle

\begin{problem}[2.50, p. 126] If $W$ is a subspace of an $n$-dimensional vector space $V$, show that $\dim(W)\leq n$. If $\dim(W)= \dim(V)$, show that $W= V$.
\end{problem}






\begin{problem}[cf. 2.64 and 2.65, p. 143]. Consider the matrix $$A = \begin{bmatrix*}[r] 2  & 1 & 3 & 1\\ 1 & 1 & 3 & 0\\ 0 & 1 & 2 & 1\\ 3 & 3 & 8 & 2\end{bmatrix*}$$
\end{problem}

\begin{itemize}
\item[a)] Decide whether $[4,1,2,5]$ and $[1,2,3,4]$ each belongs to the row space.

\item[b)] Find a basis for the row space.

\item[c)] Check that ${\rm rank}(A) = {\rm rank}(A^t)$.

\end{itemize}


\begin{problem}[2.66, p. 144] Find bases for the column and row spaces.
\end{problem}

\begin{itemize}
\item[a)] $$\begin{bmatrix*}[r] 1 & 2 & 0 & -2\\ 2 & 3 & 2 & 3\\ 2 & 10 & 4 & 10 \end{bmatrix*}$$

\item[b)] $$\begin{bmatrix*}[r] -1 & 4 & -2\\ 4 & 4 & 2\\ 3 & 0 & -3\end{bmatrix*}$$

\item[c)] $$\begin{bmatrix*}[r] 2 & 1 & 1\\1 & 3 & 2\\5 & 0 & 1\end{bmatrix*}$$

\end{itemize}


\begin{problem}[2.81, p. 146] If $A$ is an $n\times n$ matrix of rank $n$, what are the row and column spaces?
\end{problem}





\begin{problem}[5.3, p. 280] Given each $3\times 3$ matrix $A$, verify the given vectors are eigenvectors and find an expression for $A^n\vec x$ were $\vec x = [1,1,2]^t$.
\end{problem}

\begin{itemize}
\item[a)] $$A = \begin{bmatrix*}[r] 0 & 3 & -3\\ 2 & 2 & -2\\ -4 & -1 & 1\end{bmatrix*},\ \begin{bmatrix*}[r] 1\\ 1\\\ -1\end{bmatrix*},\ \begin{bmatrix*}[r] 1\\ 0\\ 1\end{bmatrix*},\ \begin{bmatrix*}[r] 0\\1\\1\end{bmatrix*}
$$

\item[b)] $$A = \begin{bmatrix*}[r] 2 & 2 & -2\\ -1 & 5 & -1\\-1 & 1 & 3\end{bmatrix*},\ \begin{bmatrix*}[r] 2\\ 1\\\ -1\end{bmatrix*},\ \begin{bmatrix*}[r] 2\\ 1\\ 1\end{bmatrix*},\ \begin{bmatrix*}[r] 1\\1\\0\end{bmatrix*}
$$


\end{itemize}

\begin{problem}[cf. 5.5 and 5.49, p. 280 and 305] Find all eigenvalues and a basis of eigenvectors for each eigenvalue.
\end{problem}

\begin{itemize}
\item[a)] $$\begin{bmatrix*}[r] -18 & 30\\-10 & 17\end{bmatrix*}$$

\item[b)] $$\begin{bmatrix*}[r] 10 & -17\\ 6 & -10\end{bmatrix*}$$

\item[c)] $$\begin{bmatrix*}[r] 1 & 2 & 0\\ -3 & 2 & 3\\ -1 & 2 & 2\end{bmatrix*}$$

\item[d)] $$\begin{bmatrix*}[r] \cos(\theta) & -\sin(\theta)\\\sin(\theta) & \cos(\theta)\end{bmatrix*}$$

\end{itemize}


\begin{problem}[cf. 5.12, p. 281] Let $p(x) = a_k x^k + a_{k-1}x^{k-1} + \dotsb + a_1 x + a_0$ be a real polynomial. If $A$ is an $n\times n$ matrix with eigenvalue $\lambda$, show that $p(\lambda)$ is an eigenvalue for the matrix $p(A) = a_k A^k + a_{k-1} A^{k-1} + \dotsb + a_1 A + a_0I$. (Hint. What should be the eigenvector?)
\end{problem}

\begin{problem} Let $A$ be a $3\times 3$ real matrix with eigenvalues $1, 2, -1$ and corresponding eigenvectors $\vec v_1$, $\vec v_2$, and $\vec v_3$. Show that $\{\vec v_1,\vec v_2, \vec v_3\}$ is a basis for $\mathbb R^3$.
\end{problem}





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