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 {\bf Name~}\underline{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}\hfill
 {Class Time~}\underline{~~~~~~~~}\newline


\begin{center}
\bf Math 2270 Extra Credit Problems \\
\bf Chapter 2 \\
{\small\bf December 2011}
\end{center}

These problems were created for Bretscher's textbook, but apply for Strang's book, except for the division by chapter.
To find the background for a problem, consult Bretscher's textbook, which can be checked out from the math library or the LCB Math Center.

{\bf Due date}: See the internet due dates.
Records are locked on that date and only corrected,
never appended.

{\bf Submitted work}. Please submit one stapled package.
Kindly label problems \fbox{\bf Extra Credit}. Label each problem with
its corresponding problem number. You may attach this printed sheet to
simplify your work.

\Prob{XC2.1-16}{(Invertible $T$)}
Decide invertibility of $T(\.x)=A\.x$ for the following matrices $A$. Then
find the matrix of $T^{-1}$, in each case.
$$\twobytwo{1&2}{3&0},\quad \twobytwo{1&1}{-1&1},\quad \twobytwo{1&1}{-1&-1}.$$


\Prob{XC2.1-43}{(Matrix of $T$)}
~\newline
(a) Suppose $\.v$ has components $2$, $-2$, $5$. Find the matrix of
$T(\.x)=\.v\cdot\.x$.
~\newline
(b) Prove that every linear transformation $T$ from $\Rr^3$ into $\Rr^1$ can
be written as $T(\.x)=\.v\cdot\.x$ for some vector $\.v$.


\Prob{XC2.1-46}{(Matrix of $T$)}
~\newline
(a) Let $T(\.x)=B(A(\.x))$ where
$$A=\twobytwo{1&2}{3&0},\quad B=\twobytwo{1&1}{-1&1}.$$
Find the matrix of $T$.
~\newline
(b) What is the matrix of $T$ defined by $T(\.x)=B^2(A(\.x))$?

\Prob{XC2.2-18}{(Reflection line equation)}
Let a reflection $T$ have matrix $\frac12\twobytwo{\sqrt3&1}{1&-\sqrt3}$.
Find the equation for the line $L$ of reflection.

\Prob{XC2.2-26}{(Matrix of $T$)}
~\newline
(a) Find the scaling matrix $A$ if $T\left(\colvector{2\\ -1}\right)=\colvector{8\\ -4}$.
~\newline
(b) Find the projection matrix $A$ if $T\left(\colvector{2\\ 3}\right)=\colvector{2\\ 0}$.

\Prob{XC2.2-39}{(Composite linear transformations)}
Each of the matrices below is a standard geometric linear transformation
followed by a scaling. Find the scale factor.
$$\twobytwo{2&2}{2&2},\quad
\twobytwo{6&0}{-2&6},\quad
\twobytwo{3/4&1}{1&-3/4}.$$


{\bf End of extra credit problems chapter 2.}

\end{document}
