\begin{problem}{MATH 294}{\spr{87}}{\pr{3}}{8}{}
\prob{Find $c_3$ so that: \[ \sqbrc{\begin{array}{c}
1 \\
0 \\
0 \\
0 \\
0
\end{array}} = c_1 \vec{v}_1 + c_2 \vec{v}_2 + c_3 \vec{v}_3 + c_4 \vec{v}_4 \] where $\vec{v}_1 = \vtfive{1}{1}{1}{1}{1}, \vec{v}_2 = \vtfive{1}{0}{-1}{0}{0}, \vec{v}_3 = \vtfive{3}{2}{3}{-4}{-4}, \vec{v}_4 =
\vtfive{0}{0}{0}{2}{-2}$}\probnn{Note that the four vectors
$\vec{v}_1, \vec{v}_2, \vec{v}_3, $ and $\vec{v}_4$ are mutually
orthogonal.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{92}}{FINAL}{6}{}
\prob{Given $A = \brc{\arthree{5}{1}{0}{1}{5}{0}{0}{0}{4}}$}
\probpart{Find an orthogonal matrix $C$ such that $C^{-1} A C$ is
diagonal. (The columns of an orthogonal matrix are orthonormal
vectors.)} \probpart{If $\vtthree{1}{1}{1} = a \vec{v}_1 + b
\vecv_2 + d \vecv_3 $} \probpartnn{where $\vecv_1, \vecv_2, $ and
$\vecv_3$ are the columns of $C$, find the scalars $a, b$ and
$d$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{FINAL}{\spr{93}}{3}{}
\prob{Consider the matrix \[ A =
\brc{\arthree{1}{0}{1}{0}{1}{1}{-1}{1}{0}.}
\]} \probpart{Find the vectors $\vec{b} = \vtthree{b_1}{b_2}{b_3}$ such that a solution $\vec{x}$ of the equation $A \vec{x} = \vec{b}$
exists.}\probpart{Find a basis for the column space
$\mathcal{R}(A)$ of $A$} \probpart{It is claim that
$\mathcal{R}(A)$ is a plane in $\Re^3$. If you agree, find a
vector $n$ in $\Re^3$ that is normal to this plane. Check your
answer.} \probpart{Show that $n$ is perpendicular to each of the
columns of $A$. Explain carefully why this is true.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\fa{94}}{\pr{3}}{5}{}
\prob{True/False} \probnn{Answer each of the following as True or
False. If False, explain, by an example. } \probpart{Every
spanning set of $\Re^3$ contain at least three vectors.}
\probpart{Every orthonormal set of vectors in $\Re^5$ is a basis
for $\Re^5$.} \probpart{Let $A$ be a 3 by 5 matrix. Nullity $A$
is at most 3.} \probpart{Let $W$ be a subspace of $\Re^4$. Every
basis of $W$ contain at least 4 vectors.} \probpart{In $\Re^n,
||cX|| = |c| ||X||$} \probpart{If $A$ is an $n \times n$
symmetric matrix, then rank $A = n.$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{97}}{\pr{3}}{4}{}
\prob{Consider $\mathcal{W}$, a subspace of $\Re^4$, defined as
$\mathcal{w}\{ \vec{v}_1, \vec{v}_2 \}$ where $\vec{v}_1 =
\vtfour{0}{-1}{1}{0}, \vecv_2 = \vtfour{1}{1}{1}{1}.$}
\probnn{$\mathcal{W}$ is a "plane" in $\Re^4.$} \probpart{Find a
basis for a subspace $\mathcal{U}$ of $\Re^4$ which is orthogonal
to $\mathcal{W}$.} \probpartnn{Hint: Find \emph{all} vectors
$\vtfour{x_1}{x_2}{x_3}{x_4}$ that are perpendicular to both
$\vecv_1$ and $\vecv_2$.} \probpart{What is the geometrical
nature of $\mathcal{U}$?} \probpart{Find the vector in
$\mathcal{W}$ that is closest to the vector $\vec{y} =
\vtfour{-1}{0}{0}{1}$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{unknown}{unknown}{?}{}
\prob{Let $W$ be the subspace of $\Re^3$ spanned by the
orthonormal set $\set{ \frac{(1,2,-1)}{\sqrt{6}},
\frac{(1,0,1)}{\sqrt{2}}}$. Let $X = (1,1,1).$ Find a vector $Z$,
in $W$, and a vector $Y$, perpendicular to every vector in $W$,
such that $X = Z+Y.$ What is the distance from $X$ to $W$?}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{99}}{\pr{3}}{1}{}
\prob{Let the functions $f_1 = 1, f_2 = t f_3 = t^2$ be three
"vectors" which span a subspace, $S$, in the vector space of
continuous functions on the interval $-1 \leq t \leq 1
(C[-1,1])$, with inner product \[ \equiv \int^1_{-1}
f(t)g(t) d t.
\] Find three orthogonal vectors, $u_1 =1, u_2 = ?, u_3 = ?$ that span $S$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING ?}{FINAL}{10}{}
\prob{Consider the vector space $C_0(-\pi, \pi)$ of continuous
functions in the interval $-\pi \leq x \leq \pi$, with inner
product conjugation. Consider the following set of functions $b =
\set{\ldots e^{-2 i x}, e^{-i x}, 1, e^{i x}, e^{2 i x}, \ldots}.
$} \probpart{Are they linearly independent? (Hint: Show that they
are orthogonal, that is} \probpartnn{$\brc{e^{i n x}, e^{i m x}} =
0$ for $n \neq m$.} \probpartnn{$\brc{e^{i n x}, e^{i m x}} \neq
0$ for $n = m$.} \probpart{Ignoring the issue of convergence for
the moment, let $f(x)$ be in $C_0(-\pi, \pi).$ Express $f(x)$ as
a linear combination of the basis $B$. That is, \[ f = \ldots
a_{-2} e^{-2 i x} + a_{-1} e^{- i x} + a_0 + a_1 e^{i x} + a_2
e^{2 i x} + \ldots
\] find the coefficients $\set{a_n}$ of each of the basis vectors. Use the results from
(a).}\probpart{How does this relate to the Fourier series? Are
the coefficients $\set{a_n}$ real or complex? What if $B$ is a
set of arbitrary orthogonal functions?}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{99}}{\pr{2}}{2a}{}
\probb{Three matrices $A,B,$ and $P$ have:} \probparts{$A = P^{-1}
B P,$} \probparts{$B$ is symmetric $(B^T = B)$, and}
\probparts{$P$ is orthogonal ($P^T = P^{-1}$).} \probnn{Is it
necessary true that $A$ is symmetric? If so, prove it. If not,
find a counter example (say three $2 \times 2$ matrices $A, B$
and $P$ where (i) - (iii) above are true and $A$ is not
symmetric).}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{99}}{\pr{3}}{4}{}
\prob{The temperature, $u(x,y),$ in a rectangular plate was
measured at six locations. The $(x,y)$ coordinates and measured
temperatures, $u$, are given in the table below. \[ \begin{array}{ccc}
x & y & u \\
0 & 0 & 11 \\
\frac{\pi}{2} & 0 & 19 \\
0 & 1 & 1 \\
\frac{\pi}{2} & 1 & 14
\end{array} \] Assume that $u(x,y)$ is supposed to obey the equation (this is \emph{not} a PDE question) \[ u(x,y) = \beta_0 + \beta_1 e^{-y} \sin x. \] Set up, but do not solve, a system of equations for the parameters, $\beta_0, \beta_1,$ that provide the least-squares best fit of the measured data to the equation
above.}\probnn{\emph{Extra credit} Neatly write out a sequence of
Matlab commands that will give you the parameters $\beta_0,
\beta_1.$ }
\end{problem}