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\lhead{Mathematics 262}
\rhead{J.~Palmieri}

\begin{center}
{\large \textbf{Exam 2}} \\
\textbf{Due by beginning of class, Wednesday, April 22}
\end{center}

General instructions: This is an \emph{open book and open notes} exam.
Of course, you can't discuss the exam with anyone except me.  You may 
not use calculators or computers.  (If you can figure out a helpful 
way to use a slide rule for any of these problems, you are welcome to 
do so.)

%Unless otherwise specified, treat all matrices (etc.)  as being 
%defined over the real numbers $\real$.

\begin{questions}

\uplevel{\textbf{Part 1.  You must do this problem.}}
	
\question
Let $O(n)$ denote the set of $n \times n$ orthogonal matrices (with 
real entries).  

\begin{parts}
\part
Show that every matrix in $O(n)$ has determinant equal to $1$ or $-1$.

\uplevel{$SO(n)$ is the subset of $O(n)$ of all orthogonal matrices 
with determinant one.  Matrices in $SO(n)$ are called ``special 
orthogonal'' matrices.}

\part
Describe the sets $SO(1)$ and $SO(2)$.  (In other words, tell me all 
of the matrices contained in each of these sets.)

\part
Suppose that $A \in SO(3)$.  Assume that $a_{13} \neq 0$.  Show that
the values of $a_{11}$, $a_{12}$, $a_{21}$, and $a_{22}$, together
with the sign of $a_{13}$, determine the rest of the entries of $A$.
(If you have this much information about a special orthogonal matrix,
explain how to find all of the entries.)

\part 
\textbf{(extra credit)} Give examples to show that you need at least
this much information (if you knew less, show that you can't uniquely
determine all of the entries of the matrix).

\part 
\textbf{(extra credit)} The set of $n \times n$ ``special unitary'' 
matrices, $SU(n)$, is the set of all unitary matrices (complex 
entries) with determinant one.  Figure out as much as possible about 
$SU(2)$.
\end{parts}




\uplevel{\textbf{Part 2---material from Chapter 6.  Do two of the 
following five problems.}}

\question
Consider the matrix $\displaystyle A = \begin{bmatrix}
0 & 0 & -c \\
1 & 0 & -b \\
0 & 1 & -a 
\end{bmatrix}$.
Show that the minimal polynomial of $A$ is $x^{3} + ax^{2} + bx + c$.  

\question
Do problem 13 in Section 6.4 (p.~206).

\question
Do problem 3 in Section 6.5 (p.~208).

%\question
%Do problem 5 in Section 6.5 (p.~208).

\question
Let $\displaystyle A = 
\begin{bmatrix} 1 & \frac{1}{2} \\ y & 3
\end{bmatrix}$.  

\begin{parts}
\part
For which complex numbers $y$ is $A$ similar (over $\complex$) to an 
upper triangular matrix?

\part
For which complex numbers $y$ is $A$ similar (over $\complex$) to a 
diagonal matrix?

\part
For which real numbers $y$ is $A$ similar (over $\real$) to a 
diagonal matrix?
\end{parts}

\question
\begin{parts}
\part
Find a $3 \times 3$ matrix $A$ so that
\begin{itemize}
\item $1$ is an eigenvalue of $A$, with eigenvector $(0,0,1)$,
\item $5$ is an eigenvalue of $A$, with eigenvector $(\frac{3}{5},
\frac{4}{5}, 0)$, and
\item $-10$ is an eigenvalue of $A$, with eigenvector $(\frac{4}{5},
-\frac{3}{5}, 0)$.
\end{itemize}

\part
Indeed, if $A$ is $n \times n$ and satisfies the equations $A \alpha_{i} 
= c_{i} \alpha_{i}$ for $i=1, \ldots, n$, tell me how to find $A$.
\end{parts}




\uplevel{\textbf{Part 3---material from Chapter 8.  Do two of the 
following three problems.}}

\question
Let $V$ denote the vector space of all polynomials with real 
coefficients, of degree at most \emph{two}; put the usual inner 
product on it:
\[
\inner{f}{g} = \int_{0}^{1} f(x) g(x) dx.
\]
Based on what we did on a homework problem, we know that if we apply 
the Gram-Schmidt procedure to the basis $\basis{1, x, x^{2}}$, we get 
this orthonormal basis:
\[
\basis{1, 2 \sqrt{3} (x-\frac{1}{2}), 6\sqrt{5} (x^{2} - x + \frac{1}{6})}.
\]
(In the homework problem, we were working with polynomials of degree 
at most three, so we had a fourth polynomial, in addition.)

\begin{parts}
\part
Apply Gram-Schmidt to the basis $\basis{x^{2}, x, 1}$ to get another
orthonormal basis for $V$ (note: not just orthogonal, but orthonormal).

\part
Use either of these orthonormal bases to find a polynomial $g(x)$ so 
that 
\[
\int_{0}^{1} f(x) g(x) dx = f'(0)
\]
for every polynomial $f \in V$.  (Hint: the function that sends a 
polynomial $f(x)$ to the number $f'(0)$ is a linear functional on 
$V$.)
\end{parts}

\question
Let $W_{1}$ and $W_{2}$ be the following subspaces of $\real^{3}$:
\begin{gather*}
W_{1} = \text{Span}((1, 1, 1), (1, 0, 1)), \\
W_{2} = \text{Span}(0, 0, 1).
\end{gather*}

\begin{parts}
\part
Show that $\textbf{R}^{3} = W_{1} \oplus W_{2}$.

\part
Find an inner product on $\real^{3}$ such that $W_{1} =
W_{2}^{\perp}$.  If you can't work out the details for some reason,
explain the procedure for solving the problem.
\end{parts}

\question
Let $\displaystyle A = 
\begin{bmatrix} a & \frac{1}{2} \\ b & - \frac{\sqrt{3}}{2} 
\end{bmatrix}$.

\begin{parts}
\part
For which complex numbers $a$ and $b$ is $A$ unitary?

\part
For which complex numbers $a$ and $b$ is $A$ self-adjoint?

\part
(\textbf{extra credit}) For which complex numbers $a$ and $b$ is $A$ normal?

\end{parts}

\end{questions}

\end{document}
