\begin{problem}{MATH 293}{\fa{95}}{\pr{1}}{3}{}
\probb{Find the orthogonal (scalar) projection of the vector
$\vec{v} = \vec{i} + \vec{j} + \vec{k}$ in the direction of the
vector $\vec{w} = 5 \vec{i} + 12 \vec{j}$} \probpart{Consider the
two vectors \[ \vec{a} = 3 \vec{i} - 4 \vec{j} \] \[ \vec{b} = 3
\vec{i} + 4 \vec{j}
\] The vector $\vec{u}$ has orthogonal projections $-\frac{1}{5}$ and $\frac{7}{5}$ along the vectors $\vec{a}$ and $\vec{b}$, respectively. Find
$\vec{u}.$} \probpartnn{Hint: Let $\vec{u} = u_1 \vec{i} + u_2
\vec{j}$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\spr{95}}{FINAL}{8}{}
\probb{What is the formula for the scalar orthogonal projection
of a vector $\vecv \in \Re^n$ onto the line spanned by a vector
$\vec{w}$.} \probpartnn{Let \[ \vec{b}_1 = \vttwo{1}{1} \mbox{
and } \vec{b}_2 = \vttwo{1}{3}.
\] Suppose $\vec{v}_1$ has orthogonal projection 3 and 7 onto the lines spanned by $\vec{b}_1$ and $\vec{b}_2$
respectively.}\probpart{Find $\vec{v}_1$.} \probpart{Suppose
$\vec{v}_2$ has orthogonal projections -6 and -14 onto the lines
spanned by $\vec{b}_1$ and $\vec{b}_2$ respectively. Find
$\vec{v}_2$.} \probpart{Are $\vec{v}_1$ and $\vecv_2$ linearly
independent.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{97}}{\pr{3}}{6}{}
\prob{\newcommand*{\sumoh}[0]{\sum_1^{100}} As part of their plan
to take over the world, lab assistant Pinky has collected 100
points of data \[ (x_1,y_1), (x_2, y_2), \ldots, (x_{100},
y_{100},
\] (which represent some devious no-good data) which his partner, Brain, will analyze. A computer program boils down this data into the following set of numbers: \[
\sumoh x_i = 10, \sumoh
x_i^2 = 20, \sumoh x_i^3 = 100, \sumoh x_i^6 = 200,
\] and \[ \ \sumoh y_i = 200, \sumoh x_i y_i = 230, \sumoh x_i^2 y_i = 250, \sumoh x_i^3 y_i = 300.\] Brain has determined that the data is probably of the form $y = a + b x^3.$ Your job is to find the least-squares solution to this problem (i.e. find the $a$ and $b$ that gives the least-squares solution).}
\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}{\fa{97}}{FINAL}{8}{}
\prob{The following figure shows numerical results $y_i,$ for $i
= 1,2, \ldots, n. $ It is known that the exact solution of the
problem is a formula of the form $y = c$, for some constant $c$.
Find the least squares solution for the constant $c$ in terms of
$y_1, y_2, \ldots, y_n,$ and $n$.$$\centerline{\epsfxsize=2.4in
\epsfbox{2_10_155.jpg}}$$}
%------------------------------------------------------------------------------(PICTURE P.155)
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{98}}{FINAL}{6}{}
\probb{Find orthonormal eigenvectors $\set{\vecv_1, vecv_2}$ of
$A$. [Hint: do not go on to parts d-e below until you have double
checked that you have found two orthogonal unit vectors that are
eigenvectors of $A$.} \probpart{Use the eigenvectors above to
diagonalize $A$.} \probpart{Make a clear sketch that shows the
standard basis vectors $\set{\vec{e}_1, \vec{e}_2}$ of $\Re^2$
and the eigenvectors $\vecv_1, \vecv_2$ of $A$.} \probpart{Give a
geometric interpretation of the change of coordinates matrix,
$P$, that maps coordinates of a vector with respect to the eigen
basis to coordinates with respect to the standard basis.}
\probpart{Let $\vec{b} = \vttwo{3}{5}$. Using orthogonal
projection express $\vec{b}$ in terms of $\set{\vecv_1, \vecv_2}$
the eigenvectors of $A$. }
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{98}}{\pr{3}}{1}{}
\prob{Consider the following three vectors in $\Re^3$: \[ \vec{y}
= \vtthree{1}{0}{1}, \vec{u}_1 = \vtthree{1}{1}{1}, \mbox{ and }
\vec{u}_2 = \vtthree{1}{-1}{0}.
\] [Note: $\vec{u}_1 \mbox{ and } \vec{u}_2$ are orthogonal.].}
\probpart{Find the orthogonal projection of $\vec{y}$ onto the
subspace of $\Re^3$ spanned by $\vec{u}_1$ and $\vec{u}_2.$}
\probpart{What is the distance between $\vec{y}$ and
$span\set{\vec{u}_1, \vec{u}_2}$?} \probpart{In terms of the
standard basis for $\Re^3$, find the matrix of the linear
transformation that orthogonally projects vectors onto
$span\set{\vec{u}_1, \vec{u}_2}.$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\fa{94}}{\pr{3}}{14}{}
\prob{The vectors $\set{(1,0,0,-1),(1,-1,0,0),(0,1,0,1)} $ are
linearly independent and span a subspace $S$ of $\Re^4$. Use the
Gram-Schmidt process to find an orthogonal basis for the subspace
of $S$ that is orthogonal to the first vector of the given set,
$(1,0,0,-1).$ }
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\fa{95}}{FINAL}{7}{}
\probb{Find an orthonormal basis for the space of vectors in
$\Re^3$ having the form $\vtthree{c_1 - c_2}{c_2}{2 c_2}$. You
may use Gram-Schmidt or any other method.} \probpart{If
$\set{\vec{b}_1, \vec{b}_2, \vec{b}_3, \vec{b}_4}$ is an
orthonormal basis for $\Re^4$, \[ \vtfour{1}{-9}{0}{\sqrt{5}} =
c_1 \vec{b}_1 + c_2 \vec{b}_2 + c_3 \vec{b}_3 + c_4 \vec{b}_4,
\mbox{ and } \vec{b}_2 = \vtfour{0.5}{0}{\alpha}{0},
\] (where $c_i$ are real constants), find the possible values of $c_2$ and $\alpha$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{97}}{\pr{3}}{1}{}
\prob{Let \[ A = \brc{\begin{array}{cccc}
1 & 1 & -1 & 1 \\
2 & 1 & 2 & 1
\end{array}}. \]} \probpart{Find an orthogonal basis for the null space of
$A$.}\probpart{Find a basis for the orthogonal complement of
$Nul(A)$, i.e. find $(Nul(A))^T$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{97}}{\pr{3}}{12}{}
\prob{Let $A = [\vec{v}_1 \vec{v}_2]$ be a $1000 \times 2$
matrix, where $\vec{v}_1, \vec{v}_2$ are the columns of $A$. You
aren't given $A$. Instead you are given only that \[ A^T A =
\brc{\artwo{1}{\frac{1}{2}}{\frac{1}{2}}{1}}.\] Find an
\textbf{orthonormal}basis $\set{\vec{u}_1, \vec{u}_2}$ of the
column space of $A$. Your formulas for $\vec{u}_1$ and
$\vec{u}_2$ should be written as linear combinations of
$\vec{v}_1, \vecv_2.$ (Hint: what do the entries of the matrix
$A^T A$ have to so with dot products?)}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SPRING ?}{FINAL}{2}{}
\probb{Find a basis for the row space of the matrix \[ A = \brc{\begin{array}{cccc}
1 & 1 & 0 & 1 \\
1 & 0 & 0 & 2 \\
0 & 0 & 4 & 0 \\
1 & 2 & 0 & 0
\end{array}}
\]} \probpart{Find the rank of $A$ and a basis for its column space, noting that $A =
A^T.$}\probpart{Construct an orthonormal basis for the row space
of $A$.}
\end{problem}