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Linear Algebra 2270-2 \\
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Due in Week 9
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The ninth  week finishes chapter 4 and starts the work from chapter 5. Here's the
list of problems, problem notes and answers.

\begin{description}
\item[{\bf Section 4.4}.] Exercises 3, 7, 9, 13, 27

\item[{\bf Section 4.5}.] Exercises 5, 7, 11, 13, 21

\item[{\bf Section 4.6}.] Exercises 1, 3, 5, 7, 15, 21

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\item[{\bf Extra Credit Problem week9-1}.] Define a function $T$ from $\Rr^n$ to $\Rr^m$ by the matrix multiply formula $T(\vec{x})=A\vec{x}$. Prove that for all vectors $\vec{u}, \vec{v}$ and all constants $c$, (a) $T(\vec{u}+\vec{v})=T(\vec{u})+T(\vec{v})$, (b) $T(c\vec{u})=cT(\vec{u})$. {\small\em Definition: $T$ is called a {\bf linear transformation} if $T$ maps $\Rr^n$ into $\Rr^m$ and satisfies (a) and (b).}

\item[{\bf Extra Credit Problem week9-2}.]  Let $T$ be a linear transformation from $\Rr^n$ into $\Rr^n$ that satisfies $\|T(\vec{x})\|=\|\vec{x}\|$ for all $\vec{x}$. Prove that the $n\times n$ matrix $A$ of $T$ is orthogonal, that is, $A^TA=I$, which means the columns of $A$ are {\bf orthonormal}:
     $$\col(A,i)\cdot\col(A,j)=0\quad\mbox{for}\quad i\ne j,\quad\mbox{and}\quad \col(A,i)\cdot\col(A,i)=1.$$

\item[{\bf Extra Credit Problem week9-3}.] Let $T$ be a linear transformation given by $n\times n$ orthogonal matrix $A$. Then $\|T(\vec{x})\|=\|\vec{x}\|$ holds. Construct an example of such a matrix $A$ for dimension $n=3$, which corresponds to holding the $z$-axis fixed and rotating the $xy$-plane $45$ degrees counter-clockwise. Draw a 3D-figure which shows the action of $T$ on the unit cube $S=\{ (x,y,z) ~:~ 0\le x \le 1, 0\le y\le 1, 0\le z \le 1\}$.

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\subsection*{Problem Notes}

{\bf Extra Credit Problem week9-1}. Write out both sides of identities (a) and (b), replacing $T(\vec{w})$ by matrix product $A\vec{w}$ for various choices of $\vec{w}$. Then compare sides to finish the proof.

{\bf Extra Credit Problem week9-2}. Equation $\|T(\vec{x})\|=\|\vec{x}\|$ means lengths are preserved by $T$. It also means $\|A\vec{x}\|=\|\vec{x}\|$, which applied to $\vec{x}=\col(I,k)$ means $\col(A,k)$ has length equal to $\col(I,k)$ (=1).
Write $\|\vec{w}\|^2=\vec{w}\cdot\vec{w}=\vec{w}^T\vec{w}$ (the latter a matrix product). Then write out the equation
$\|A\vec{x}\|^2=\|\vec{x}\|^2$, to see what you get, for various choices of unit vectors $\vec{x}$.

{\bf Extra Credit Problem week9-3}. The equations for such a transformation can be written as plane rotation equations in $x$, $y$ plus the
identity in $z$. They might look like $x'=x\cos\theta -y\sin\theta$, $y'=$ similar, $z'=z$. Choose $\theta$ then test it by seeing what happens to $x=1$, $y=0$, $z=0$, the answer for which is a rotation of vector $(1,0,0)$. The answer for $A$ is obtained by writing the scalar equations as a matrix equation $(x',y',z')^T=A(x,y,z)^T$.

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