\begin{problem}{MATH 294}{FALL 1981}{PRELIM 1}{3}{}
\probb{Show that the set of vectors $$ \{1+t, 1-t,1-t^2 \} $$ is a
basis for the vector space of all polynomials $$ \vec{p} = a_0 +
a_1 t + a_2 t^2 $$ of degree less than three.} \probpart{Express
the vector $$2 + 3 t + 4 t^2$$ in terms of the above basis.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1982}{PRELIM 1}{2}{}
\prob{Let V be the space of all solutions of $$ \vec{x} = \left[
\begin{array}{ccc}
  0 & 0 & 1 \\
  0 & 1 & 0 \\
  1 & 0 & 0
\end{array}\right] \vec{x}. $$ Consider the vectors $$ \vec{x}_1 (t) = \left( \begin{array}{c}
  e^{-t} \\
  0 \\
  -e^{-t}
\end{array}\right), \vec{x}_2 (t) = \left( \begin{array}{c}
  e^{t} \\
  0 \\
  e^{t}
\end{array}\right).$$}
\probpart{Do $ \vec{x}_1(t)$, $\vec{x}_2(t)$ belong to $V$?}
\probpart{Are $ \vec{x}_1(t)$, $\vec{x}_2(t)$ linearly
independent? Give reasons for your answer.} \probpart{Do the
vectors $ \vec{x}_1(t)$, $\vec{x}_2(t)$ form a basis for $V$? Give
reasons for your answer.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1983}{FINAL}{10}{}
\probb{Find a basis for the vector space of all $2 \times 2$
matrices.} \probpart{$A$ is the matrix given below, $ \vec{v}$ is
an eigenvector of A.  Find any eigenvalue of $A$. $A = \left[
\begin{array}{cccc}
  3 & 0 & 4 & 2 \\
  8 & 5 & 1 & 3 \\
  4 & 0 & 9 & 8 \\
  2 & 0 & 1 & 6
\end{array}\right]$ with $\vec{v} =$ [an eigenvector of A] $= \left( \begin{array}{c}
  0 \\
  2 \\
  0 \\
  0
\end{array}\right)$}
\probpart{Find one solution to each system of equations below, if
possible.  If not possible, explain why not. $\left[
\begin{array}{cccc}
  1 & 1 & 1 & 1 \\
  2 & 2 & 2 & 2 \\
  3 & 3 & 3 & 3 \\
  4 & 4 & 4 & 4
\end{array}\right] \cdot \vec{x} = \vec{b}$ , $\vec{b} = \left[ \begin{array}{c}
  1 \\
  2 \\
  3 \\
  4
\end{array} \right]$ and $\vec{b} = \left[ \begin{array}{c}
  1 \\
  0 \\
  0 \\
  0
\end{array} \right]$ }
\probpart{Read carefully.  Solve for $\vec{x}$ in the equation $A
\cdot \vec{b} = \vec{x}$ with: $A = \left[ \begin{array}{ccc}
  1 & 2 & 3 \\
  3 & 2 & 1 \\
  1 & 0 & 1
\end{array} \right]$ and  $\vec{b} = \left[ \begin{array}{c}
  1 \\
  0 \\
  1
\end{array} \right].$}
\probpart{Find the inverse of the matrix $A = \left[ \begin{array}{ccc}
  1 & 0 & 0 \\
  0 & 1 & 0 \\
  0 & 0 & 1
\end{array} \right].$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1984}{FINAL}{2}{}
\prob{Determine whether the given vectors form a basis for $S$,
and find the dimension of the subspace.  $S$ is the set of all
vectors of the form $(a, b, 2a, 2b)$ in $\Re^4$.  The given set is
$\{(1,0,2,0),(0,1,0,3),(1,-1,2,-3)\}$. }
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1986}{FINAL}{1}{}
\prob{The vectors
(1,0,2,-1,3),(0,1,-1,2,4),(-1,1,-2,1,-3),(0,1,1,-2,-4), and
(1,4,2,-1,3) span a subspace $S$ of $\Re^5$.} \probpart{What is
the dimension of $S$?} \probpart{Find a basis for $S$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1986}{FINAL}{2}{}
\probb{Solve the linear system $A \vec{x} = \vec{b}$, where $A =
\left[ \begin{array}{cccc}
  1 & 0 & -2 & 4 \\
  2 & 1 & -4 & 6 \\
  -1 & 2 & 5 & -3 \\
  3& 3 & -5 & 4
\end{array} \right]$ and $\vec{b} = \left[ \begin{array}{c}
  4 \\
  9 \\
  9 \\
  15
\end{array}\right]$.}
\probpart{Solve the linear system $A \vec{x} = \vec{0}$, where $A
= \left[ \begin{array}{ccccc}
  -3 & -1 &  0 &  1 & -2 \\
   1 &  2 & -1 &  0 &  3 \\
   2 &  1 &  1 & -2 &  1 \\
   1 &  5 &  2 & -5 &  4
\end{array} \right]$ Express your answer in vector form, and give a basis for the space of solutions.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1987}{PRELIM 3}{6}{}
\prob{Find an \underline{orthonormal} basis for the subspace of
$\Re^3$ consisting of all 3-vectors $\left( \begin{array}{c}
  x \\
  y \\
  z
\end{array} \right)$ such that $x+y+z=0$. }
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1989}{PRELIM 3}{3}{}
\prob{Let $W$ be the following subspace of $\Re^3$, $$W = Comb
\left( \left[ \begin{array}{c}
  1 \\
  0 \\
  1
\end{array} \right], \left[ \begin{array}{c}
  1 \\
  1 \\
  -1
\end{array} \right], \left[ \begin{array}{c}
  2 \\
  1 \\
  0
\end{array} \right], \left[ \begin{array}{c}
  3 \\
  3 \\
  -3
\end{array} \right] \right)$$}
\probpart{Show that $\left[ \begin{array}{c}
  1 \\
  0 \\
  1
\end{array} \right],\left[ \begin{array}{c}
  1 \\
  1 \\
  -1
\end{array} \right],$ is a basis for $W$}
\probnn{For b) and c) below, let $T$ be the following linear
transformation $T:W\rightarrow\Re^3$. $$T\left( \left[ \begin{array}{c}
  w_1 \\
  w_2 \\
  w_3
\end{array} \right]
\right) = \left[ \begin{array}{ccc}
  1 & 0 & -1 \\
  0 & 0 & 0 \\
  0 & 0 & 0
\end{array} \right] \left[ \begin{array}{c}
  w_1 \\
  w_2 \\
  w_3
\end{array} \right]$$ for those $\left[ \begin{array}{c}
  w_1 \\
  w_2 \\
  w_3
\end{array} \right]$ in $\Re^3$ which belong to $W$.
[ You are allowed to use a) even if you did not solve it.] }
\probpart{What is the dimension of Range($T$)? (Complete
reasoning, please.)} \probpart{What is the dimension of Ker($T$)?
(Complete reasoning, please.)}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SPRING 1990}{PRELIM 1}{3}{}
\prob{Find the dimension and a basis for the following spaces}
\probpart{The space spanned by
\{$(1,0,-2,1)$,$(0,3,1,-1)$,$(2,3,-3,1)$,$(3,0,-6,-1)$\}}
\probpart{The set of all polynomials $p(t)$ in $P^3$ satisfying
the two conditions} \probparts{$\frac{d^3}{dt^3}p(t)=0$ for all t}
\probparts{$p(t) + \frac{d}{d t}p(t)=0$ at $t=0$} \probpart{The
subspace of the space of functions of t spanned by $\left\{ e^{a
t}, e^{b t} \right\}$ if $ a \neq b$.} \probpart{The space spanned
by $\left\{ \vec{v}_1,\vec{v}_2,\vec{v}_3,\vec{v}_4 \right\}$ in
$W$, given that  $\left\{ \vec{v}_2,\vec{v}_3,\vec{v}_4 \right\}$
is a basis for $W$. }
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SPRING 1990}{PRELIM 1}{4}{}
\probb{Show that $B = \left\{ t^2-1,t^2+1,t \right\}$ is a basis
for $P^2$} \probpart{Express the vectors in $\left\{1,t,t^2
\right\}$ in terms of those in $B$ and find the components of
$p(t) = (1+t)^2$ with respect to $B$.}  \probpart{Find the
components of the vector $\vec{x} = (1,2,3)$ with respect to the
basis \{(1,0,0),(1,1,0),(1,1,1)\}.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{FALL 1990}{PRELIM 2}{1}{}
\probb{Express the vectors $\vec{u},\vec{v}$ in terms of
$\vec{a},\vec{b}$, given that
$$3\vec{u}+2\vec{v}=\vec{a},\vec{u}-\vec{v}=\vec{b}$$}
\probpart{If $\vec{a},\vec{b}$ are linearly independent, find a
basis for the span of \{ $\vec{u},\vec{v}$,$\vec{a},\vec{b}$ \}}
\probpart{Find $\vec{u},\vec{v}$, if $\vec{a} = (-1,2,8),\vec{b} =
(-2,-1,1)$ }
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{FALL 1991}{PRELIM 3}{1}{}
\prob{Consider the matrix $$A = \left( \begin{array}{cccc}
   2 & -1 &  1 &  3 \\
  -1 &  2 & -2 & -2 \\
   2 &  5 & -4 &  1 \\
   1 &  4 & -4 &  0
\end{array}\right)$$} \probpart{Find a basis for the row space of
$A$.} \probpart{Find a basis for the column space of $A$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SPRING 1992}{PRELIM 3}{6}{}
\prob{Given $A = \left( \begin{array}{cccc}
  1 &  0 & 2 & 3 \\
  0 &  1 & 1 & 2 \\
  1 &  1 & 3 & 5 \\
  2 & -1 & 3 & 4
\end{array} \right)$. } \probpart{Find a basis for the null space of
$A$.} \probpart{Find the rank of $A$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SUMMER 1992}{PRELIM 7/21}{3}{}
\prob{Given a matrix $A = \left( \begin{array}{cccc}
  1 &  0 & 1 & -1 \\
  0 &  2 & 1 &  2 \\
  1 &  2 & 2 &  1 \\
  1 & -2 & 0 & -3
\end{array} \right)$. } \probpart{Find a basis for the row space $W_1$ of $A$.
}\probpart{Find a basis for the range $W_2$ of $A$.}
\probpart{Find the rank of $A$.} \probpart{Are the two space $W_1$
and $W_2$ the same subspace of $V_4$? Explain your answer
carefully in order to get credit for this part. }
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SPRING 1992}{FINAL}{2}{}
\probb{Find a basis for $V_4$ that contains at least two of the
following vectors: $$ \vec{v}_1 = \left( \begin{array}{c}
   1 \\
   0 \\
   1 \\
  -1
\end{array} \right), \vec{v}_2 = \left( \begin{array}{c}
   0 \\
   1 \\
   1 \\
   1
\end{array} \right),\vec{v}_3 = \left( \begin{array}{c}
   1 \\
   1 \\
   2 \\
   0
\end{array} \right) $$} \probpart{$A$ is a $3\times3$ matrix. If $A \left( \begin{array}{c}
  1 \\
  1 \\
  3
\end{array} \right) = \left( \begin{array}{c}
  0 \\
  4 \\
  7
\end{array} \right)$ and $ \left\{ \left( \begin{array}{c}
  1 \\
  0 \\
  0
\end{array} \right), \left( \begin{array}{c}
  1 \\
  1 \\
  0
\end{array} \right) \right\}$ is a basis for the nullspace of $A$, then find the general solution $\vec{x}$ of the equation $A \vec{x} = \left( \begin{array}{c}
  0 \\
  4 \\
  7
\end{array} \right)$.  Find, also, the determinant of $A$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SUMMER 1992}{PRELIM 7/21}{4}{}
\prob{Given four vectors in $V_4$ $$ \vec{v}_1 = \left(
\begin{array}{c}
   2 \\
   4 \\
  -2 \\
  -4
\end{array} \right), \vec{v}_2 = \left( \begin{array}{c}
   1 \\
   2 \\
  -1 \\
  -2
\end{array} \right),\vec{v}_3 = \left( \begin{array}{c}
   4 \\
   4 \\
   0 \\
  -6
\end{array} \right) ,\vec{v}_4 = \left( \begin{array}{c}
   1 \\
   0 \\
   1 \\
  -1
\end{array} \right)$$} \probpart{Find the space $W$ spanned by the vectors ($\vec{v}_1,\vec{v}_2,\vec{v}_3,\vec{v}_4 $)
}\probpart{Find a basis for $W$.} \probpart{Find a basis for $V_4$
that contains as many of the vectors
$\vec{v}_1$,$\vec{v}_2$,$\vec{v}_3$ and $\vec{v}_4$ as possible.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{FALL 1992}{PRELIM 3}{2}{}
\prob{Consider the matrix $$A = \left( \begin{array}{cccc}
   1 &  1 & -1 &  1 \\
   0 &  1 &  2 &  2 \\
   2 &  0 & -6 & -2 \\
  -1 &  1 &  5 &  3
\end{array} \right)$$} \probpart{Find a basis for the column space of $A$ from among the set of column
vectors.}\probpart{Find a basis for the row space of $A$.}
\probpart{Find a basis for the null space of $A$.} \probpart{What
is the rank of $A$ and the dimension of the null space (the
nullity)? }
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{FALL 1992}{PRELIM 3}{3}{}
\prob{Let $C(-\pi,\pi)$ be the vector space of continuous
functions on the interval $-\pi \leq x \leq \pi$.  Which of the
following subsets $S$ of $C(-\pi,\pi)$ are subspaces? If it is not
a subspace say why.  If it is, then say why and find a basis.}
\probnn{Note: You must show that the basis you choose consists of
linearly independent vectors.  In what follows $a_0$, $a_1$ and
$a_2$ are arbitrary scalars unless otherwise stated.}
\probpart{$S$ is the set of functions of the form $f(x) = 1 +
a_1\sin(x) + a_2\cos(x)$} \probpart{$S$ is the set of functions of
the form $f(x) = 1 + a_1\sin(x) + a_2\cos(x)$, subject to the
condition $ \int^{\pi}_{-\pi} f(x)\,\mathrm{d}x = 2 \pi $}
\probpart{$S$ is the set of functions of the form $f(x) = 1 +
a_1\sin(x) + a_2\cos(x)$, subject to the condition $
\int^{\pi}_{-\pi} f(x)\,\mathrm{d}x = 0 $}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{FALL 1992}{FINAL}{3}{}
\probb{Let $A$ be an $n \times n$ nonsingular matrix.  Prove that
$ \det(A^{-1}) = \frac{1}{\det(A)} $.  Hint: You may use the fact
that if $A$ and $B$ are $n \times n$ matrices $\det (A B) =
\det(A) \det(B)$. } \probpart{An $n \times n$ matrix $A$ has a
nontrivial null space.  Find $\det(A)$ and explain your answer.}
\probpart{Given two vectors $\vec{v}_1 = \left( \begin{array}{c}
   1 \\
   1 \\
   1
\end{array} \right)$ and $\vec{v}_2 = \left( \begin{array}{c}
   1 \\
   0 \\
   1
\end{array} \right)$ in $V_3$. Find a vector (or vectors) $\vec{w}_1$,$\vec{w}_2$,... in $V_3$ such that the set \{ $\vec{v}_1$,$\vec{v}_2$,$\vec{w}_1$,... \} is a basis for $V_3$.}
\probpart{Let $S$ be the set of all vectors of the form $\vec{v} =
a \vec{i} + b\vec{j} +c\vec{k}$ where $\vec{i}$, $\vec{j}$ and
$\vec{k}$ are the usual mutually perpendicular unit vectors.  Let
$W$ be the set of all vectors that are perpendicular to the vector
$\vec{v} = \vec{i} + \vec{j} + \vec{k}$. Is $W$ a vector subspace
of $V_3$? Explain your answer. }
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SPRING 1993}{PRELIM 3}{2}{}
\prob{Given the matrix $B = \left( \begin{array}{cccc}
  1 & 2 & 3 & 4 \\
  2 & 3 & 5 & 7 \\
  0 & 1 & 2 & 3 \\
  3 & 3 & 4 & 5
\end{array} \right) $} \probpart{Find a basis for the row space of $B$} \probpart{Find a basis for the null space of $B$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SPRING 1993}{PRELIM 3}{14}{}
\prob{Consider the following vectors in $\Re^4$}
\probnn{$\vec{v}_1 = \left( \begin{array}{c}
   1 \\
   0 \\
  -1 \\
   1
\end{array} \right)$, $\vec{v}_2 =
\left( \begin{array}{c}
   2 \\
  -3 \\
  -8 \\
   2
\end{array} \right)$,$\vec{v}_3 =
\left( \begin{array}{c}
   0 \\
   1 \\
   2 \\
   0
\end{array} \right)$,$\vec{v}_4 =
\left( \begin{array}{c}
   3 \\
   1 \\
  -1 \\
   3
\end{array} \right)$} \probnn{Let $W$ be the subspace of $\Re^4$ spanned by the vectors $\vec{v}_1$,$\vec{v}_2$,$\vec{v}_3$ and $\vec{v}_4$.}
\probnn{Find a basis for $W$ which is contained in (is a subset
of) the set \{ $\vec{v}_1$,$\vec{v}_2$,$\vec{v}_3$, $\vec{v}_4$
.\}}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SPRING 1993}{PRELIM 3}{5}{}
\probb{Consider the vector space $V$ whose elements are $3 \times
3$ matrices.} \probparts{Find a basis for the subspace $W_1$ of
$V$ which consists of all upper-triangular $3 \times 3$ matrices.}
\probparts{Find a basis for the subspace $W_1$ of $V$ which
consists of all upper-triangular $3 \times 3$ matrices with zero
trace.} \probpartnnn{The trace of a matrix is the sum of its
diagonal elements.} \probpart{Consider the polynomial space $P^3$
of polynomials with degree $\leq 3$ on $0\leq t \leq 1$.}
\probpartnn{Find a basis for the subspace $W$ of $P^3$ which
consists of polynomials of degree $\leq 3$ with the constraint $$
\left[ \frac{d^2p}{dt^2} + \frac{d p}{d t} \right]_{t=0} = 0.$$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{FALL 1994}{PRELIM 3}{1}{}
\prob{Let $A$ be the matrix $\left[ \begin{array}{cccc}
  1 & 2 & -1 &  3 \\
  2 & 2 & -1 &  2 \\
  1 & 0 &  0 & -1
\end{array} \right]$} \probpart{Find a basis for the Null Space of $A$.  What is the nullity of
$A$?}\probpart{Find a basis for the Row Space of $A$.  What is its
dimension? } \probpart{Find a basis for the Column Space of $A$.
What is its dimension? } \probpart{What is the rank of $A$?}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{FALL 1994}{FINAL}{4}{}
\probb{Find a basis for the space spanned by:
\{(1,0,1),(1,1,0),(-1,-4,-3)\}. } \probpart{Show that the
functions $e^{2 x}\cos(x)$ and $e^{2 x}\sin(x)$ are linearly
independent.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SPRING 1995}{PRELIM 3}{3}{}
\prob{Let $P_3$ be the space of polynomials $p(t)$ of degree $\leq
3$.  Consider the subspace $S \subset P_3$ of polynomials that
satisfy $$ p(0) + \frac{d p}{d t} \, \bigg | _{t=0} = 0$$ }
\probpart{Show that $S$ is a subspace of $P_3$.}  \probpart{Find a
basis for $S$.} \probpart{What is the dimension of $S$?}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SPRING 1995}{PRELIM 3}{5}{}
\probb{Find a basis for the plane $P \subset \Re^3$ of equation $$
x + 2 y + 3 z = 0$$ } \probpart{Find an orthonormal basis for
$P$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{FALL 1995}{PRELIM 3}{5}{}
\prob{Let $P_3$ be the space of polynomials $p(t) = a_0 + a_1 t +
a_2 t^2 + a_3 t^3$ of degree $\leq 3$. Consider the subset $S$ of
polynomials that satisfy $$p ^{\prime\prime}(0) = 4 p(0) = 0 $$ }
\probnn{Here $p ^{\prime \prime} (0) $ means, as usual,
$\frac{d^2p}{d t^2} \Big| _{t=0}$.} \probpart{Show that $S$ is a
subspace of $P_3$.  Give reasons.} \probpart{Find a basis for
$S$.} \probpart{What is the dimension of $S$?  Give reasons for
your answer.} \probnn{Hint: What constraint, if any, does the
given formula impose on the constants $a_0, a_1, a_2,$ and $a_3$
of a general $p(t)$?}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{FALL 1995}{FINAL}{2}{}
\prob{Consider the subspace $W$ of $\Re^4$ which is defined as $$
W = span \left\{ \left[ \begin{array}{c}
   0 \\
  -1 \\
   1 \\
   0
\end{array} \right], \left[ \begin{array}{c}
   1 \\
  -1 \\
   0 \\
   1
\end{array} \right] \right\}$$ } \probpart{Find a basis for $W$.}
\probpart{What is the dimension of $W$?} \probpart{It is claimed
that $W$ is a ``plane" in $\Re^4$.  Do you agree?  Give reasons
for your answer. } \probpart{It is claimed that the ``plane" $W$
can be described as the intersection of two 3-D regions $S-1$ and
$S_2$ in $\Re^4$.  The equations of $S-1$ and $S_2$ are:$$ \begin{array}{cc}
  {S_1:} & x-u=0 \\
  {S_2:} & a x + b y + c z + d u = 0
\end{array} $$ }
\probpartnn{where $ \left[ \begin{array}{c}
  x \\
  y \\
  z \\
  u
\end{array} \right] $ is a generic point in $\Re^4$ and $a, b,c ,d$ are real
constants.}\probpartnn{Find one possible set of values for the
constants $a,b,c,$ and $d$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SPRING 1996}{PRELIM 3}{1}{}
\prob{The set $W$ of vectors in $\Re^3$ of the form $(a,b,c)$,
where $a+b+c=0$, is a subspace of $\Re^3$.} \probpart{Verify that
the sum of any two vectors in $W$ is again in $W$.} \probpart{The
set of vectors $$S = {(1,-1,0),(1,1,-2),(-1,1,0),(1,2,-3)}$$ is in
$W$.  Show that $S$ is linearly dependent.} \probpart{Find a
subset of $S$ which is a basis for $W$.} \probpart{If the
condition $a+b+c=0$ above is replaced with $a+b+c=1$, is $W$ still
a subspace? Why/ why not? }
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SPRING 1996}{PRELIM 3}{2}{}
\prob{Which of the following subsets are bases for $\Re^2$? Show
any algebra involved or state a theorem to justify your answer. $$
S_1 = \left\{ \left[ \begin{array}{c}
  1 \\
  0
\end{array} \right], \left[ \begin{array}{c}
  0 \\
  1
\end{array} \right], \left[ \begin{array}{c}
  1 \\
  1
\end{array} \right] \right\}, S_2 = \left\{ \left[ \begin{array}{c}
  1 \\
  2
\end{array} \right], \left[ \begin{array}{c}
  3 \\
  4
\end{array} \right] \right\}, S_3 = \left\{ \left[ \begin{array}{c}
  1 \\
  2
\end{array} \right], \left[ \begin{array}{c}
  -3 \\
  -6
\end{array} \right] \right\}.$$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SPRING 1996}{FINAL}{22}{}
\prob{Let $$W = Span \left\{ \left[ \begin{array}{c}
  1 \\
  1 \\
  1
\end{array} \right], \left[ \begin{array}{c}
  \frac{1}{3} \\
  \frac{1}{3} \\
  -\frac{2}{3}
\end{array} \right] \right\}.$$} \probnn{Then an orthonormal basis for $W$
is}\probpart{$\left\{ \left[ \begin{array}{c}
  \frac{1}{\sqrt{3}} \\
  \frac{1}{\sqrt{3}} \\
  \frac{1}{\sqrt{3}}
\end{array} \right], \left[ \begin{array}{c}
  \frac{1}{3} \\
  \frac{1}{3} \\
  -\frac{2}{3}
\end{array} \right] \right\}$} \probpart{$\left\{ \left[ \begin{array}{c}
  1 \\
  1 \\
  1
\end{array} \right], \left[ \begin{array}{c}
  1 \\
  -2 \\
  1
\end{array} \right] \right\}$}
\probpart{$\left\{ \left[ \begin{array}{c}
  \frac{1}{\sqrt{3}} \\
  \frac{1}{\sqrt{3}} \\
  \frac{1}{\sqrt{3}}
\end{array} \right], \left[ \begin{array}{c}
  \frac{1}{\sqrt{6}} \\
  -\frac{2}{\sqrt{6}} \\
  \frac{1}{\sqrt{6}}
\end{array} \right] \right\}$}
\probpart{$\left\{ \left[ \begin{array}{c}
  \frac{1}{\sqrt{3}} \\
  \frac{1}{\sqrt{3}} \\
  \frac{1}{\sqrt{3}}
\end{array} \right], \left[ \begin{array}{c}
  \frac{1}{\sqrt{6}} \\
  \frac{1}{\sqrt{6}} \\
  -\frac{2}{\sqrt{6}}
\end{array} \right] \right\}$} \probpart{none of the above}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1997}{PRELIM 2}{2}{}
\prob{Consider the vector space $P_2$ of all polynomials of degree
$\le 2$.  Consider two bases of $P_2$:}
\probpartnn{$S:\{1,t,t^2\}$, the standard basis, and}
\probpartnn{$H: \{1,2 t, -2+4 t^2\}$, the Hermite basis.}
\probpart{Find the matrices $P_{S \leftarrow H}$ and $P_{H
\leftarrow S}$.} \probpart{Consider $p_1(t) = 1 + 2 t + 3 t^2$ in
$P_2$, and $p_2(t) = \frac{d}{d t}p_1(t)$. Find $$ [p_1(t)]_S,
[p_2(t)]_S, [p_1(t)]_H, [p_2(t)]_H, $$ i.e. the coordinates of
$p_1$ and $p_2$ in the bases $S$ and $H$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1997}{PRELIM 2}{3}{}
\prob{Let $W$ be the subspace of $\Re^4$ defined as $$ W = span
\left( \left( \begin{array}{c}
   1 \\
   1 \\
  -2 \\
   0
\end{array} \right), \left( \begin{array}{c}
   1 \\
   1 \\
   0 \\
  -2
\end{array} \right),\left( \begin{array}{c}
   1 \\
   1 \\
  -6 \\
   4
\end{array} \right)\right).$$} \probpart{Find a basis for $W$.  What is the dimension of
$W$?}\probpart{It is claimed that $W$ can be described as the
intersection of two linear spaces $S_1$ and $S_2$ in $\Re^4$. The
equations of $S_1$ and $S_2$ are $$S_1: x-y =0,$$ and $$S_2: a x +
b y + c z + d w = 0,$$ where $a,b,c,d$ are real constants that
must be determined.  Find one possible set of values of $a,b,c$
and $d$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1997}{PRELIM 2}{6}{}
\prob{Let $V$ be the vector space of $2 \times 2$ matrices.}
\probpart{Find a basis for $V$.} \probpart{Determine whether the
following subsets of $V$ are subspaces.  If so, find a basis.  If
not, explain why not.} \probparts{ \{ $A$ in $V \big| \det A = 0
$\} } \probparts{ \{ $A$ in $V \big| A \left( \begin{array}{c}
  0 \\
  1
\end{array} \right) = A \left( \begin{array}{c}
  1 \\
  0
\end{array} \right) $\}. } \probpart{Determine whether the following are linear transformations. Give a short justification for your answers.}
\probparts{$T: V \rightarrow V,$ where $T(A) = A^T$,}
\probparts{$T: V \rightarrow \Re^1,$ where $T(A) = \det A$,}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1998}{FINAL}{4}{}
\prob{Here we consider the vector spaces $P_1$, $P_2$, and $P_3$
(the spaces of polynomials of degree 1,2 and 3).} \probpart{Which
of the following transformations are linear? (Justify your
answer.) } \probparts{$T: P_1 \rightarrow P_3, T(p) \equiv t^2p(t)
+ p(0)$} \probparts{$T: P_1 \rightarrow P_1, T(p) \equiv p(t) +
t$} \probpart{Consider the linear transformation $T: P_2
\rightarrow P_2$ defined by $T(a_0 + a_1 t + a_2t^2) \equiv (-a_1
+ a_2) + (-a_0 + a_1)t + (a_2)t^2$. With respect to the standard
basis of $P_2$,  $B = \{1,t,t^2\}$, is $A = \left[
\begin{array}{ccc}
   0 & -1 & 1 \\
  -1 &  1 & 0 \\
   0 &  0 & 1
\end{array} \right]$. Note that an eigenvalue/eigenvector pair of $A$ is $\lambda = 1, \vec{v} = \left[ \begin{array}{c}
  0 \\
  1 \\
  1
\end{array} \right]$. Find an eigenvalue/eigenvector (or eigenfunction) pair of $T$.  That is, find $\lambda$ and $g(t)$ in $P_2$ such that $T ( g(t) ) = \lambda g(t)$.
} \probpart{Is the set of vectors in $P_2 \{3+t,-2+t,1+t^2\}$ a
basis of $P_2$? (Justify your answer.)}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SPRING ?}{FINAL}{C}{}
\prob{Give a definition for addition and for scalar multiplication
which will turn the set of all pairs $(\vec{u}, \vec{v})$ of
vectors, for $\vec{u}, \vec{v}$ in $V_2$, into a vector space
$V$.} \probpart{What is the zero vector of $V$?} \probpart{What is
the dimension of $V$? } \probpart{What is a basis for $V$?}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{87}}{\pr{3}}{2 MAKE-UP}{}
\prob{On parts (a) - (g), answer true or false.}
\probpart{$span\brc{\vv_1, \vv_2, \vv_3, \vv_4} = \Re^3$, where
$\vv_1 = \vtthree{1}{2}{3}, \vv_2 = \vtthree{3}{2}{1}, \vv_3 =
\vtthree{1}{0}{-1}, \vv_4 = \vtthree{0}{1}{1}.$} \probpart{The
four vectors in (a) are independent.} \probpart{Referring to a
again, all vectors $\vv = \vtthree{x_1}{x_2}{x_3}$ in
$span\brc{\vv_1, \vv_2, \vv_3, \vv_4}$ satisfy a linear equation
$a x_1 + b x_2 + c x_3 = 0$ for scalars a,b,c not all 0.}
\probpart{The rank of the matrix $\brc{\begin{array}{ccc}
  1 & 2 & 3 \\
  3 & 2 & 1 \\
  1 & 0 & -1 \\
  0 & 1 & 1
\end{array}}$ is 3.} \probpart{In $Re^n$ n distinct vectors are
independent.}\probpart{$n+1$ distinct vectors always span
$\Re^n$, for $n>1.$} \probpart{If the vectors $\vv_1, \vv_2,
\ldots , \vv_n$ span $\Re^n$, then they are a basis for $\Re^n$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{UNKNOWN}{PRACTICE}{4a}{}
\probb{Find a basis for the row space of the matrix \[ A =
\sqbrc{\begin{array}{cccc}
   1 &  2 & -1 &  4 \\
   3 &  6 &  1 & 12 \\
   9 & 18 &  1 & 36
\end{array}}\]}
\end{problem}
%----------------------------------
\begin{problem}{UNKNOWN}{UNKNOWN}{UNKNOWN}{?}{}
\prob{If $A$ is an $m \times n$ matrix show that $B = A^T A \and
C = A A^T $ are both square. What are their sizes? Show that $B =
B^T, C = C^T$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL ?}{FINAL}{1 MAKE-UP}{}
\prob{Consider the homogeneous system of equations $B \vec{x} =
\vec{0} $, where \[ B = \sqbrc{\begin{array}{rrrrr}
  0 &  1 & 0 & -3 & 1 \\
  2 & -1 & 0 &  3 & 0 \\
  2 & -3 & 0 &  0 & 4
\end{array}}, \; \vec{x} = \vtfive{x_1}{x_2}{x_3}{x_4}{x_5}, \and \; \vec{0} =
\vtthree{0}{0}{0}
\]}
\probpart{Find a basis for the subspace $W \subset \Re^5,$ where
$W = $ set of all solutions of $B \vec{x} = \vec{0} $}
\probpart{Is B 1-1 (as a transformation of $\Re^5 \to \Re^3$)?
Why?} \probpart{Is B:$\Re^5 \to \Re^3$ onto why?} \probpart{Is
the set of all solutions of $B \vec{x} = \vtthree{3}{0}{0}$ a
subspace of $\Re^5?$ Why?}
\end{problem}
