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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 2.7}.] Exercises 2, 7, 20, 40

\item[{\bf Section 3.1}.] Exercises 1, 3, 11, 12, 17, 18, 22, 23 

\item[{\bf Section 3.2}.] Exercises 4, 10, 15, 17, 18, 19, 24, 37


\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 2.7}. Exercises 2, 20 are 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
$\twobytwo{\cos\theta &-ô\sin\theta}{\sin\theta & \cos\theta}$.

{\bf 3.1}. Exercises 1, 3, 11, 23 are in the textbook.

{\bf 3.1-12}. For the plane $x+y-2z=4$, the sum of $(4,0,0)$ and $(0,4,0)$
is not on the plane. (The key is that this plane does not go through $(0,0,0)$.)

{\bf 3.1-17}.  (a) The invertible matrices do not include the zero
matrix, so they are not a subspace

(b) The sum of singular matrices
$\twobytwo{1& 0}{0& 0}+\twobytwo{0& 0}{ 0& 1}$
is not singular: not a subspace.

{\bf 3.1-18}. (a) True: The symmetric matrices do form a subspace (b)
True: The matrices with $A^T=-A$ do form a subspace (c) False: The sum
of two unsymmetric matrices could be symmetric.


{\bf 3.1-22}. (a) Solution for every b (b) Solvable only if $b_3=0$. (c)
Solvable only if $b_3=b_2$.


{\bf 3.2}. Exercises  4, 15, 24, 37 are in the textbook. 
10, 17, 18, 19,



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