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\usepackage{mathtools}
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\begin{document}

\title{MATH 351, Fall 2017, Homework \#3}

\author{\underline{\bf Due Friday, September 15}}



\maketitle


\begin{problem} True or false? Explain.
\end{problem}

\begin{itemize}
\item[a)] The row vectors of a matrix in echelon form are linearly independent as long as each row vector is non-zero.

\item[b)] The column vectors of a matrix in echelon form are linearly independent as long as each column vector is non-zero. 

\end{itemize}


\begin{problem} Find a map $T: \mathbb R^2 \to \mathbb R^2$ so that $T(\vec x + \vec y) = T(\vec x) + T(\vec y)$ for all $\vec x, \vec y\in \mathbb R^2$ but which is not linear.
\end{problem}

\begin{problem} Find a map $T: \mathbb R^2 \to \mathbb R^2$ so that $T(c\cdot\vec x) = c\cdot T(\vec x) $ for all $\vec x\in \mathbb R^2$ and all $c\in \mathbb R$ but which is not linear.
\end{problem}

\begin{problem} Find a map $T: \mathbb C^2 \to \mathbb C^2$ so that $T(c\cdot\vec x) = c\cdot T(\vec x)$ for all $\vec x\in \mathbb C^2$ and all $c\in \mathbb R$ but which is not linear. (We are regarding $\mathbb C^2$ as a vector space over $\mathbb C$.)
\end{problem}


\begin{problem} Let $W_1$ and $W_2$ be subspaces of the same vector space $V$. We define the set $W_1+W_2$ as follows: $$W_1 + W_2 = \{\vec x + \vec y : \vec x\in W_1\ \textup{and}\ \vec y\in W_2\}.$$ Show that $W_1+W_2$ is a subspace of $V$.
\end{problem}

\begin{problem} Let $W_1$ and $W_2$ be subspaces of the same vector space $V$. We define the set $W_1\cap W_2$ as follows: $$W_1 \cap W_2 = \{\vec x : \vec x\in W_1\ \textup{and}\ \vec x\in W_2\}.$$ Show that $W_1\cap W_2$ is a subspace of $V$.

\end{problem}

\noindent What about $W_1\cup W_2 = \{\vec x : \vec x\in W_1\ \textup{or}\ \vec x\in W_2\}$?


\begin{problem} Find the null space of each of the following matrices.
\end{problem}

\begin{itemize}

\item[a)] $$A = \begin{bmatrix*}[r] 1 & 1 & -1\\ 1 & -1 & 1\\ -1 & 1 & 1\end{bmatrix*}$$
%The command {bmatrix*}[r] produces a right-aligned, rather than center aligned, matrix array so that the negative entries look nice

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

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

\end{itemize}


\begin{problem} Find a vector $\vec b$ such that $A\vec x = \vec b$ is inconsistent if one exists.
\end{problem}


\begin{itemize}

\item[a)] $$A = \begin{bmatrix*}[r] 1 & 1 & -1\\ 1 & -1 & 1\\ -1 & 1 & 1\end{bmatrix*}$$

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

\item[c)] $$A = \begin{bmatrix*}[r] 1 & 1 & -5 & 8 & 7\\ -1 & 2 & 0 & -7 & 6\\ 0 & 3 & 10 & 0 & 0\\ 0 & -6 & -5 & -1 & -13\end{bmatrix*}$$

\end{itemize}

\end{document}


