\documentclass[11pt]{article}
\usepackage{linalg}
\usepackage[pdftex]{graphicx}
\parindent=0in
\parskip=1ex
\begin{document}

\begin{center}\large\bf
Linear Algebra 2270-2 \\
\normalsize\rm
Week 3
\end{center}

The third week finishes the work from chapter 2. Here's the
list of problems, followed by problem notes and a few answers.

\begin{description}
\item[{\bf Section 2.1}.] Exercises 9, 10, 21, 29, 34

\item[{\bf Section 2.2}.] Exercises 5, 9, 12, 13, 32

\item[{\bf Section 2.3}.] Exercises 3, 13, 19, 25, 27, 28, 29

\end{description}


\subsection*{Problem Notes}

Resolution of issues for Strang's problems will appear here in response to communications. If there is a difficulty, like a missing definition or incomplete information, then please write email, call 581-6879, or visit JWB 113.

{\bf 1.2-13}.

{\bf 1.3-6}.

{\bf 1.3-10}.

\subsection*{Some Answers}

Only answers not in the back of the book appear below.

{\bf 2.1-9}. (a) $Ax = (18, 5, 0)$ and (b) $Ax = (3, 4, 5, 5)$.

{\bf 2.1-10}. Multiplying as linear combinations of the columns gives the same $Ax$. By rows or by
columns: 9 separate multiplications for 3 by 3.

{\bf 2.1-21}. $R=\frac{1}{\sqrt2}\left(\begin{array}{ll} 1 & -1 \\ 1 & 1 \end{array}\right)$. The entries are sines and cosines of $45$ degrees.

{\bf 2.1-29}.  $u_2 =\colvector{.7 \\ .3}$ and $u_3=\colvector{.65\\ .35}$. The components add to 1. They are always positive.
$u_7, v_7, w_7$ are all close to $\colvector{6.\\ .4}$. Their components still add to $1$.

{\bf 2.1-34}. $\displaystyle A=\left(\begin{array}{rrrr} 2 & -1 & 0 & 0 \\
                                          -1 & 2 & -1 & 0 \\
                                           0 & -1 & 2 & -1 \\
                                           0 &  0 & -1 & 2
                 \end{array}\right)$ and $A\vec{x}=\colvector{1\\ 2\\ 3\\ 4}$
                has solution $\vec{x}=\colvector{4\\ 7\\ 8 \\ 6}$.

{\bf 2.2-5}.
$6x + 4y$ is 2 times $3x + 2y$. There is no solution unless the right
side is $2\times 10 = 20$. Then all the points on the line $3x+2y = 10$
are solutions, including $(0, 5)$ and $(4,-1)$. (The two lines in the row
picture are the same line, containing all solutions).






{\bf 2.2-9}.

{\bf 2.2-12}.

{\bf 2.2-13}.

{\bf 2.2-32}.


2.3-3, 13, 19, 25, 27, 28, 29


\end{document}
