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


\begin{center}
\bf Math 2270 Extra Credit Problems \\
\bf Chapter 4 \\
{\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{XC4.1-20}{(Matrix space basis)}
Find a basis and the dimension for the space of all $2\times 2$ matrices
$\twobytwo{a & b}{a+b & a-b}$.

\Prob{XC4.1-26}{(Polynomial space basis)}
Find a basis and the dimension for the space of all polynomials
$p(x)=a_0+a_1x+a_2x^2+a_3x^3$ satisfying the conditions
$p(-1)=p(1)$, $\int_0^1 p(x)dx=p(0)$.

\Prob{XC4.1-38}{(Commutator dimension)}
Find the dimension of the space of all
$3\times 3$ matrices $A$ that commute with a $3\times 3$ diagonal matrix $B$.

\Prob{XC4.1-54}{(Subspace dimension)}
Let $V$ be a vector space of dimension $n$ and let $S$ be a subspace of $V$.
Prove that $S$ has a basis of $k$ elements and $k\le n$.

\Prob{XC4.2-13}{(Linear transformations and isomorphisms)}
Let $T(A)=PA-AP$ where $P=\twobytwo{2&3}{5&7}$. Prove that $T$ is a linear
transformation on the set $V$ of all $2\times 2$ matrices $A$. Determine the
kernel and image of $T$ and report whether $T$ is an isomorphism.

\Prob{XC4.2-20}{(Kernel of a linear transformation)}
Let $T(y)=3y''+5y'$ be defined on the vector space $V$ of all twice
continuously differentiable functions $y(x)$ defined on
$-\infty<x<\infty$ with range in some vector space $W$.
Compute the kernel of $T$ and therefore show that
$T$ is not an isomorphism. How should $W$ be defined?
Is there a problem with $W=V$?

Hint: {\small\rm Write $v=y'$ and break the equation $3y''+5y'=0$ into $3v'+5v=0$
and $y'=v$. These are elementary first order differential equations. The
general solution of $3v'+5v=0$ is $v=v_0e^{-5x/3}$.}

\Prob{XC4.2-34}{(Sequence spaces)}
Let $T(f)$ denote the sequence $f(0)$, $f'(0)$, \ldots of derivatives of
$f$ evaluated at $0$. Let $V$ be the space of all polynomials. Then the
vector space $W$ is the set of all sequences with only finitely many
nonzero terms. Prove that $W$ is a subspace of the vector space $E$ of
all real sequences equipped with termwise addition and scalar
multiplication. Prove that $T$ is linear and decide if it is an
isomorphism.

\Prob{XC4.3-2}{(Independence in matrix spaces)}
Prove or disprove: a list of $2\times 2$ matrices of length $5$ or more
is dependent.

\Prob{XC4.3-30}{(Tangents)}
Let $T(p)=p(0)+xp'(0)$ be defined on the polynomials
$p(x)=a_0+a_x+a_2x^2$ of degree $2$ or less, a vector space $V$. Prove
that $T$ is linear and compute the matrix of $T$ relative to the basis
$1$, $x-1$, $(x-1)^2$.

\clearpage
\Prob{XC4.3-60}{(Matrix of change of basis)}
Let a plane $P$ be given in $\Rr^3$ having two different bases
$\{\.v_1,\.v_2\}$
and
$\{\.w_1,\.w_2\}$.
Give general formulas for the change of basis matrices $V$ and $W$ which
change one basis into the other (in both possible ways). Then give
a formula which relates $V$ to
$\AUG(\.v_1,\.v_2)$
and
$\AUG(\.w_1,\.w_2)$.


\Prob{XC4.3-64}{(Matrix of a linear transformation)}
Let $T(A)=PA-AP$ where $P=\twobytwo{2&3}{0&7}$. Prove that $T$ is a linear
transformation on the set $V$ of all $2\times 2$ upper triangular
matrices $A$. (a) Find the standard basis ${\cal B}$ of $V$. (b) Find the
matrix of $T$ relative to basis ${\cal B}$. (c) Determine the rank of $T$.



{\bf End of extra credit problems chapter 4.}

\end{document}
