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\begin{center}\large\bf
Linear Algebra 2270-2 \\
\normalsize\rm
Week 5
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The fourth 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 3.1}.] Exercises 1, 2, 3a, 10, 12, 17, 18, 22, 23 

\item[{\bf Section 3.2}.] Exercises 4,9,15,17,18,19,37

\item[{\bf Section 3.3}.] Exercises 1,12,21

\item[{\bf Section 3.4}.] Exercises 1,3,4,6,13,16,33

\item[{\bf Section 3.5}.] Exercises 2,5,9,10,11,15,18,23,26 (a and b only),31 

\item[{\bf Section 3.6}.] Exercises 1,2,5,7,11,24

\item[{\bf Section 4.1}.] Exercises 5,9,11,12,16,17,19,20,21,26

\end{description}


\subsection*{Problem Notes}

Issues for Strang's problems will be communicated here. If there is a difficulty or impasse, then please send email, call 581-6879, or visit JWB 113.


\subsection*{Some Answers}



{\bf 2.4-all}. All listed exercises have answers in the textbook.

{\bf 2.5-40}. The inverse of
$\displaystyle A=\left(\begin{array}{rrrr}
  1 & -a &0 & 0 \\
  0 & 1  &-b & 0 \\
  0 & 0 & 1 & -c \\
  0 & 0 & 0 & 1
\end{array}\right)$
is $A^{-1} =
\displaystyle \left(\begin{array}{rrrr}
  1 & a & ab & abc \\
  0 & 1  &b & bc \\
  0 & 0 & 1 & c \\
  0 & 0 & 0 & 1
\end{array}\right)$. See also the cofactor formula for the inverse in 5.3.


{\bf 2.6-all}. All answers appear in the textbook.


{\bf 2.7-7}. (a) False, (b) False, (c) True, (d) True. Supply reasons and proofs.

{\bf 2.7-40}. Start from $Q^TQ=I$, as in $\colvector{q_1^T \\ q_2^T}\rowvector{q_1,q_2} = \twobytwo{1&0}{0&1}$.

(a) The diagonal entries give $q_1^T q_1 = 1$ and $q_2^Tq_2=1$, unit vectors.

(b) The off-diagonal entry is $q_1^Tq_2=0$ (and in general $q_i^Tq_j=0$).

(c) The leading example for $Q$ is the rotation matrix
$\twobytwo{\cos\theta &-ô\sin\theta}{\sin\theta & \cos\theta}$.

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