\begin{problem}{MATH 294}{FALL 1982}{PRELIM 1}{3a}{}
\prob{Let $C[0,1]$ denote the space of continuous functions
defined on the interval [0,1] (i.e. $f(x)$ is a member of $C[0,1]$
if $f(x)$ is continuous for $0 \le x \le 1$).  Which one of the
following subsets of $C[0,1]$ does {\bfseries not} form a vector
space? Find it and explain why it does not.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1982}{PRELIM 1}{3}{}
\probb{} \probparts{The subset of functions $f$ which belongs to
$C[0,1]$ for which $\int^1_0 f(s) d s = 0$.} \probparts{The set of
functions $f$ in $C[0,1]$ which vanish at exactly one point (i.e.
$f(x) = 0$ for only one $x$ with $0 \le x \le 1$). Note different
functions may vanish at different points within the interval. }
\probparts{The subset of functions $f$ in $C[0,1]$ for which $f(0)
= f(1)$.} \probpart{Let $f(x) = x^3 + 2 x + 5$. Consider the four
vectors $\vec{v}_1 = f(x)$, $\vec{v}_2 = f^{\prime}(x)$
,$\vec{v}_3 = f^{\prime \prime}(x)$ , $\vec{v}_4 = f^{\prime
\prime \prime}(x)$, $f^{\prime}$ means $\frac{d f}{d x}$.}
\probparts{What is the dimension of the space spanned by the
vectors? Justify your answer.} \probparts{Express $x^2+1$ as a
linear combination of the $\vec{v}_i$'s.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1984}{FINAL}{1}{}
\probb{Determine which of the following subsets are subspaces of
the indicated vector spaces, and for each subspace determine the
dimension of th- space. Explain your answer, giving proofs or
counterexamples.} \probparts{The set of all vectors in $\Re^2$
with first component equal to 2.} \probparts{The set of all
vectors $\vec{x} = (x_1,x_2,x_3)$ in $\Re^3$ for which $x_1 + x_2
+ x_3 = 0 $.} \probparts{The set of all vectors in $\Re^3$
satisfying $x_1^2 + x_2^2 - x_3^2 = 0$.} \probparts{The set of all
functions $f(x)$ in $C[0,1]$ such that $\int^1_0 f(x) d x = 0$.
Recall that $C[0,1]$ denotes the space of all real valued
continuous functions defined on the closed interval [0,1].}
\probpart{Find the equation of the plane passing through the
points (0,1,0), (1,1,0) and (1,0,1), and find a unit vector normal
to this plane.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1985}{FINAL}{5}{}
\prob{Vectors $\vec{f}$ and $\vec{g}$ both lie in $\Re^n$.  The
vector $\vec{h} = \vec{f} + \vec{g}$} \probpart{Also lies in
$\Re^n$.} \probpart{May or may not lie in $\Re^n$.} \probpart{Lies
in $\Re^{\frac{n}{2}}$.} \probpart{Does not lie in $\Re^n$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1985}{FINAL}{6}{}
\prob{The vector space $\Re^n$} \probpart{Contains the zero
vector.} \probpart{May or may not contain the zero vector.}
\probpart{Never contains the zero vector.} \probpart{Is a complex
vector space.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1985}{FINAL}{7}{}
\prob{Any set of vectors which span a vector space}
\probpart{Always contains a subset of vectors which form a basis
for that space.} \probpart{May or may not contain a subset of
vectors which form a basis for that space.} \probpart{Is a
linearly independent set.} \probpart{Form an orthonormal basis for
the space.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1987}{PRELIM 3}{7}{}
\prob{For what values of the constant $a$ are the functions
\{$\sin t$ and $\sin(t+a)$\} in $C^1_\infty$ linearly
independent?}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1987}{PRELIM 3}{9}{}
\prob{For problems (a) - (c) use the bases $B$ and $B^{\prime}$
below:} \probnn{$B = \left\{ \left( \begin{array}{c}
  1 \\
  0
\end{array} \right), \left( \begin{array}{c}
  1 \\
  -1
\end{array} \right) \right\}$ and $B^{\prime} = \left\{ \left( \begin{array}{c}
  1 \\
  1
\end{array} \right), \left( \begin{array}{c}
  0 \\
  1
\end{array} \right) \right\}$.} \probpart{Given that $[\vec{v}]_B=\left( \begin{array}{c}
  2 \\
  3
\end{array} \right)$ what is $[\vec{v}]_{B^{\prime}}$?}
\probpart{Using the standard relation between $\Re^2$ and points
on the plane make a sketch with the point $\vec{v}$ clearly
marked.  Also mark the point $\vec{w}$, where $[\vec{w}]_B =
\left( \begin{array}{c}
  0 \\
  -1
\end{array} \right)$. } \probpart{Draw the line defined by the points $\vec{v}$ and $\vec{w}$.  Do the points on this line represent a subspace of $\Re^2$?}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1987}{PRELIM 3}{2}{}
\prob{In parts (a) - (g) answer ``true" if $V$ is a vector space
and ``false" if it is not (no partial credit):} \probpart{$V =$
set of all $x(t)$ in $C_\infty$ such that $x(0)=0$.} \probpart{$V
=$ set of all $x(t)$ in $C_\infty$ such that $x(0)=1$.}
\probpart{$V =$ set of all $x(t)$ in $C_\infty$ such that
$(D+1)x(t)=0$.} \probpart{$V =$ set of all $x(t)$ in $C_\infty$
such that $(D+1)x(t)=e^t$.} \probpart{$V =$ set of all polynomials
of degree less than or equal to one with real coefficients.}
\probpart{$V =$ set of all rational numbers (a rational number can
be written as the ratio of two integers, e.g., $\frac{4}{17}$ is a
rational number while $\pi = 3.14\ldots$ is not)} \probpart{$V = $
set of all rational numbers with the added restriction that
scalars must also be rational numbers.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1987}{FINAL}{7}{}
\prob{Consider the boundary-value problem} \probnn{$X^{\prime
\prime} + \lambda X = 0$ $0 < x < \pi$, $X(0) = X(\pi) = 0$, where
$\lambda$ is a given real number.} \probpart{Is the set of all
solutions of this problem a subspace of $C_{\infty}[0,\pi]$? Why?}
\probpart{Let $W =$ set of all functions $X(x)$ in
$C_{\infty}[0,\pi]$ such that $X(0) = X(\pi) = 0$.  Is $T \equiv
D^2 - \lambda$ linear as a transformation of $W$ into
$C_{\infty}[0,\pi]$? Why?} \probpart{For what values of $\lambda$
is $Ker(T)$ nontrivial? } \probpart{Choose one of those values of
$\lambda$ and determine $Ker(T)$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SPRING 1990?}{PRELIM 2}{3}{}
\prob{Is the set of vectors $ \left\{ \vec{v}_1 = e^{-t} ,
\vec{v}_2 = e^t \right\} $ in $C^{\infty}$ linearly independent or
dependent? (Justify your answer.)}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SPRING 1992}{PRELIM 2}{3}{}
\prob{$W$ is the subspace of $V_4$ spanned by the vectors $\left(
\begin{array}{c}
  1 \\
  0 \\
  2 \\
  0
\end{array} \right),$ $\left(
\begin{array}{c}
  1 \\
  1 \\
  2 \\
  0
\end{array} \right),$ $\left(
\begin{array}{c}
  0 \\
  1 \\
  0 \\
  0
\end{array} \right)$.  Find the dimension of $W$ and give a basis.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SPRING 1992}{PRELIM 2}{4}{}
\prob{$V$ is the vector space consisting of vector-valued
functions $\vec{x}(t) = \left( \begin{array}{c}
  x_1(t) \\
  x_2(t)
\end{array} \right)$ where $x_1(t)$ and $x_2(t)$ are continuous functions of $t$ in $0 \le t \le 1$. $W$ is the subset of $V$ where the functions satisfy the differential equations} \probnn{$\frac{d x_1}{d t} = x_1 + x_2$ and $\frac{d x_2}{d t} = x_1 -
x_2$}\probnn{Is $W$ a subspace of $V$?}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SPRING 1992}{PRELIM 2}{6}{}
\prob{$V$ is the vector space consisting of all $2 \times 2$
matrices $A = \left[ \begin{array}{cc}
  a_{11} & a_{12} \\
  a_{21} & a_{22}
\end{array} \right]$.  Here the $a_{ij}$ are arbitrary real numbers and the addition and scalar multiplication are defined
by}\probnn{$A+B = \left[ \begin{array}{cc}
  a_{11}+b_{11} & a_{12}+b_{12} \\
  a_{21}+b_{21} & a_{22}+b_{22}
\end{array} \right]$ and $c A = \left[ \begin{array}{cc}
  c a_{11} & c a_{12} \\
  c a_{21} & c a_{22}
\end{array} \right]$} \probpart{Is $W_1 = \left\{ all \left[ \begin{array}{cc}
  a_{11} & a_{12} \\
  0 & 1
\end{array} \right] \right\}$ a subspace?  If so give a basis for
$W_1$.}\probpart{Same as part (a) for $W_2 = \left\{ \left[
\begin{array}{cc}
  a_{11} & -a_{12} \\
  a_{12} & a_{11}
\end{array} \right] \right\}$.} \probpart{Show that $\left( \begin{array}{cc}
  1 & 3 \\
  0 & 0
\end{array} \right)$, $\left( \begin{array}{cc}
  0 & 0 \\
  1 & 0
\end{array} \right)$, and $\left( \begin{array}{cc}
  0 & 0 \\
  0 & 2
\end{array} \right)$ are linearly independent.} \probpart{What is the largest possible number of linearly independent vectors in $V$?}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SPRING 1992}{FINAL}{7}{}
\prob{A ``plane" in $V_4$ means, by definition, the set of all
points of the form $\vec{u}+\vec{x}$ where $\vec{u}$ is a constant
(fixed) vector and $\vec{x}$ varies over a fixed two-dimensional
subspace of $V_4$.  Two planes are ``parallel" if their subspaces
are the same.  It is claimed that the two planes: }
\probpartnn{$1^{st}$ plane: $$ \left( \begin{array}{c}
  1 \\
  0 \\
  0 \\
  0
\end{array} \right) + \left( \begin{array}{c}
  0 \\
  x_2 \\
  x_3 \\
  0
\end{array} \right) $$} \probpartnn{$2^{nd}$ plane: $$ \left( \begin{array}{c}
  2 \\
  0 \\
  0 \\
  0
\end{array} \right) + \left( \begin{array}{c}
  0 \\
  x_2 \\
  0 \\
  x_4
\end{array} \right) $$} \probnn{(where $x_2$, $x_3$ and $x_4$ can assume any scalar values) do not intersect and are not parallel.  Do you agree or disagree with this claim?  You have to give very clear reasons for your answer in order to get credit for this problem.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SUMMER 1992}{FINAL}{3}{}
\probb{Let $V$ be the vector space of all $2 \time 2$ matrices of
the form $\left( \begin{array}{cc}
  a_{11} & a_{12} \\
  a_{21} & a_{22}
\end{array} \right)$ } \probpartnn{where $a_{ij}$ , $i,j = 1,2,$ are real
scalars.}\probpartnn{Consider the set $S$ of all $2 times 2$
matrices of the form $\left( \begin{array}{cc}
  a+b & a-b \\
  b & a
\end{array} \right)$ } \probpartnn{where $a$ and $b$ are real
scalars.}\probparts{Show that $S$ is a subspace.  Call it $W$.}
\probparts{Find a basis for $W$ and the dimension of $W$.}
\probpart{Consider the vector space $V$ $\left\{ f(t) = a + b \sin
t + c \cos t \right\}$, for all real scalars $a$, $b$ and $c$ and
$ 0 \le t \le 1$} \probpartnn{Now consider a subspace $W$ of $V$
in which $ \frac{d f(t)}{d t} + f(t) = 0$ at $t =0 $}
\probpartnn{Find a basis for the subspace $W$.}
\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 $-\pi \le x \le \pi$.  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 $\int^{\pi}_{-\pi} f(x) d x = 2 \pi$} \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 $\int^{\pi}_{-\pi} f(x) d x = 0$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{FALL 1992}{PRELIM 2}{5}{}
\prob{Consider all polynomials of degree $\le 3$ $$ P_3 = \{ p(t)
= a_0 + a_1 t + a_2 t^2 + a_3 t^3\}, -\infty < t < \infty $$ They
Form a vector space.  Now consider the subset $S$ of $P_3$
consisting of polynomials of degree $\le 3$ with the conditions $$
p(0) = 0, \frac{d p}{d t}(0) = 0 $$ Is $S$ a subspace $W$ of
$P_3$? Carefully explain your answer.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{FALL 1992}{PRELIM 2}{6}{}
\prob{Given a vector space $V_4$ which is the space of all vectors
of the form $\left( \begin{array}{c}
  x_1 \\
  x_2 \\
  x_3 \\
  x_4
\end{array} \right)$ for all real $x_1$, $x_2$, $x_3$, $x_4$,}
\probnn{consider the set $S$ of vectors in $V_4$ of the form $$ S
= \left\{ a \left( \begin{array}{c}
  1 \\
  0 \\
  2 \\
  1
\end{array} \right) + b \left( \begin{array}{c}
  2 \\
  1 \\
  3 \\
  -2
\end{array} \right) + c \left( \begin{array}{c}
  1 \\
  0 \\
  -2 \\
  1
\end{array} \right) \right\} $$ for all values of scalars $a$, $b$ and
$c$.}\probnn{Is the set $S$ a subspace $W$ of $V_4$? Explain your
answer carefully. }
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{FALL 1992}{FINAL}{3d}{}
\prob{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
$\vec{v}_1 =  \vec{i} + \vec{j} + \vec{k}$.  Is $W$ a vector
subspace of $V_3$? Explain your answer.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{FALL 1994}{PRELIM 2}{5}{}
\prob{In each of the following, you are given a vector space $V$
and a subset $W$. Decide whether $W$ is a subspace of $V$, and
prove that your answer is correct.} \probpart{$V$ is the space
$M_{2,2}$ of all $2 \times 2$ matrices, and $W$ is the set of $2
\times 2$ matrices A such that $A^2 = A$} \probpart{$V$ is the
space of differentiable functions, and $W$ is the set of those
differentiable functions that satisfy $f^{\prime}(3) = 0$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{FALL 1994}{PRELIM 2}{4}{}
\probb{Let $M$ denote the set of ordered triples $(x,y,z)$ of real
numbers with the operations of addition and multiplication v=by
scalars $c$ defined by $$
(x,y,z)\oplus(x^{\prime},y^{\prime},z^{\prime}) =
(x+z^{\prime},y+y^{\prime},z+z^{\prime}) $$  $$ c \odot (x,y,z) =
(2 c, c y, c z). $$ Is $M$ a vector space? Why?}
\probpart{Consider the vector space $\Re^4$.  Is the subset $S$ of
vectors of the form $(x_1,x_2,x_3,x_4)$ where $x_1$, $x_2$, and
$x_3$ are arbitrary and $x_4 \le 0$ a subspace? Why?}
\probpart{Consider the vector space $P_2$ of polynomials of degree
$\le 2$.  Is the subset $S$ of polynomials of the form $p(t) = a_0
+ a_1 t + (a_0 + a_1) t^2$ a subspace? Why?}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{FALL 1994}{PRELIM 3}{5}{}
\prob{Answer each of the following as True or False.  If false,
explain, by an example.} \probpart{Every spanning set of $\Re^3$
contains 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 $W$ contains at least 4
vectors.} \probpart{In $\Re^n, ||c X|| = |c|\, ||X||$}
\probpart{If $A$ is an $n \times n$ symmetric matrix, then rank $A
= n$.}
\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}{FALL 1994}{PRELIM 3}{2}{}
\prob{Which of the following sets of vectors is linearly
independent? Show all work.} \probpart{In $P_2$: $S=\{1,t,t^2\}$}
\probpart{In $\Re^3$: $S = \{(1,2,-1),(6,3,0),(4,-1,2)\}$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SPRING 1995}{PRELIM 3}{3}{}
\prob{Let $P_3$ be the space of polynomials $p(t)$ of degree $\le
3$.  Consider the subspace $S \subset P_3$ of polynomials that
satisfy $$ p(0) + \left. \frac{d p}{d t} \right|_{t=0} =0 $$}
\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}{FALL 1995}{PRELIM 3}{3}{}
\prob{Let $P_3$ be the space of polynomials $p(t) = a_0 + a_1 t +
a_2 t + a_3 t^3$ of degree $\le 3$. Consider the subset $S$ of
polynomials that satisfy $$ p^{\prime \prime}(0) + 4 p(0) = 0 $$
Here $ p^{\prime \prime}(0)$ means, as usual, $\left.
\frac{d^2p}{d t^2} \right|_{t=0}$.} \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}{PRELIM 3}{4}{}
\prob{We define a new way of ``adding" vectors by $$ \left(
\begin{array}{c}
  x_1 \\
  x_2
\end{array} \right) + \left(
\begin{array}{c}
  y_1 \\
  y_2
\end{array} \right) = \left(
\begin{array}{c}
  x_1+y_1 \\
  x_2 y_2
\end{array} \right)$$ and use ordinary scalar multiplication.}
\probpart{Is the commutative axiom ``$x+y = y+x$" satisfied? }
\probpart{Is the associative axiom ``$x+(y+z)=(x+y)+z$"
satisfied?} \probpart{How about the distributive law ``$a (x+y)=a
x + a y$"? } \probpart{Is this a vector space?} \probnn{Give
reasons.}
\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}{3}{}
\prob{Which of the following subsets are bases for $P_2$, the
vector space of polynomials of degree less than or equal to two?
You do {\itshape not} need to show your work.} \probpartnn{$S_1 =
\{ 1 , t, 1- t, 1+t\}$, $S_2 = \{ t^2, t^2+2, t^2 + 2 t\}$, $S_3 =
\{1 + t + t^2, t,t^2\}$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SPRING 1996}{FINAL}{4}{}
\prob{Suppose $\vec{v}_1, \ldots , \vec{v}_p$ are vectors in
$\Re^n$.  Then $Span\{ \vec{v}_1, \ldots , \vec{v}_p \}$ is
always:} \probpart{a linearly independent set of vectors}
\probpart{a linearly dependent set of vectors} \probpart{a basis
for a subspace of $\Re^n$} \probpart{the set of all possible
linear combinations of $\vec{v}_1, \vec{v}_2, \ldots ,
\vec{v}_p$,} \probpart{none of the above.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1997}{FINAL}{7.2}{}
\prob{Which of the following subsets are subspaces of the vector
space $P_2$ of polynomials of degree $\le 2$? (No Justification is
necessary.) Express your answer as e.g.: SUBSPACE: a,b,c,d; NOT:
e} \probpart{\{ $p(t) \big| p^{\prime}(t) = 0$, all t\}}
\probpart{\{ $p(t) \big| p^{\prime}(t)-1 = 0$, all t\}}
\probpart{\{ $p(t) \big| p(0) + p(1) = 0$\}} \probpart{\{ $p(t)
\big| p(0) = 0$ and $p(1) = 0$ \}} \probpart{\{ $p(t) \big| p(0) =
0$ and $p(1) = 1$ \}}
\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
equation 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}{SPRING 1998}{PRELIM 3}{5}{}
\prob{True or False? Justify each answer.} \probpart{In general,
if a finite set $S$ of nonzero vectors spans a vector space $V$,
then some subset of $S$ is a basis of $V$.} \probpart{A linearly
independent set in a subspace $H$ is a basis for $H$.}
\probpart{An $n \times n$ matrix $A$ is diagonalizable if and only
if $A$ has $n$ eigenvalues, counting multiplicities. }
\probpart{If an $n \times n$ matrix $A$ is diagonalizable, it is
invertible.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SPRING ?}{FINAL}{6}{}
\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 293}{FALL 1998}{PRELIM 2}{2}{}
\prob{Given the matrix $$ A = \left[ \begin{array}{ccc}
  1 & 1 & 1 \\
  1 & -1 & 2 \\
  1 & 1 & 4
\end{array} \right], $$} \probpart{Show by a calculation that its determinant is
nonzero.}\probpart{Calculate its inverse by any means.}
\end{problem}
%----------------------------------