\begin{problem}{MATH 294}{SPRING 1982}{PRELIM 1}{3}{}
\probb{Let $C[0,1]$ denote the space of continuous function
defined on the interval $[0,1]$ (i.e. $f(x)$ is a member of
$C[0,1]$ if $f(x)$ is continuous for $0\leq x \leq 1)$.  Which
one of the following subsets of $C[0,1]$ does \textbf{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 $\int^1_0f(s)ds=0$.} \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\leq x \leq 1 $).}
\probpartnnn{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 $f(x) = x^3 +
2x + 5$. Consider the four vector $v_1=f(x), v_2=f^{\prime}
(x),v_3=f^{\prime \prime} ,v_4=f^{\prime \prime \prime} (x),
(f^{\prime} (x)$ means$\frac{df}{dx}$)} \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 $v_i$'s}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1983}{PRELIM 1}{2}{}
\prob{Consider the system $$ \left. \begin{array}{ccccccccc}
  x  & + & y  & - & z  & + & w  & = & 0 \\
  x  &   &    & + & 3z & + & w  & = & 0 \\
  2x & + & y  & + & 2z & + & 2w & = & 0 \\
  3x & + & 2y & + & z  & + & 3w & = & 0
\end{array} \right\} $$} \probpart{Find a basis for the vector space of solutions to the system above.  You need not prove this is a basis}
\probpart{What is the dimension of the vector space of solutions
above? Give a reason.} \probpart{Is the vector \[ \left[
\begin{array}{c}
  x \\
  y \\
  z \\
  w
\end{array}\right]=\left[ \begin{array}{c}
  -2 \\
  1 \\
  1 \\
  2
\end{array}\right]
\] a solution to the above system?}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1987}{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 $(a,b,2a,3b$ in $R^4$.  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}
\probnn{$(1,4,2,-1,3)$ span a subspace   $S$ of $R^5$. }
\probpart{What is the dimension of $S$? } \probpart{Find a basis
for$S$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1986}{FINAL}{1}{}
\prob{Compute the rank of the matrix \[ \left( \begin{array}{cccc}
  1 & 1 & 1 & 0 \\
  2 & 1 & 2 & 0 \\
  0 & 0 & 0 & 0 \\
  0 & 2 & 1 & -1 \\
  1 & 0 & 0 & 1
\end{array} \right) \]}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1987}{PRELIM 3}{3}{}
\prob{Find the dimension of the subspace of $R^6$ consisting of
all linear combinations of the vectors \[ \left[
\begin{array}{c}
  1 \\
  2 \\
  3 \\
  4 \\
  5 \\
  6
\end{array}
\right], \left[
\begin{array}{c}
  2 \\
  3 \\
  4 \\
  5 \\
  6 \\
  1
\end{array}
\right] , \left[
\begin{array}{c}
  1 \\
  1 \\
  1 \\
  1 \\
  1 \\
  2
\end{array}
\right] , \left[
\begin{array}{c}
  3 \\
  4 \\
  5 \\
  6 \\
  7 \\
  8
\end{array}
\right]
\]}
\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{$p(t) + frac{d p(t)}{dt} = 0$ at $t=0$} \probpart{The
subspace of the space of functions of $t$ spanned by $\left\{
e^{at},e^{bt}\right\}$ 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 2}{1}{}
\prob{Let $A = \left[ \begin{array}{cccc}
  1  & 3  & 5  & -1 \\
  -1 & -2 & -5 & 4 \\
  0  & 1  & 1  & -1 \\
  1  & 4  & 6  & -2
\end{array} \right]$  Find a basis for the \textbf{column space} of $A$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SPRING 1990}{PRELIM 2}{2}{}
\prob{Consider the equation} \probnn{$Ax = b$ ; $A = \left[
\begin{array}{cccc}
  1  & 3  & 5  & -1 \\
  -1 & -2 & -5 & 4 \\
  0  & 1  & 1  & -1 \\
  1  & 4  & 6  & -2
\end{array} \right]$} \probpart{Solve for x given $\vec{b}=\left( \begin{array}{c}
  1 \\
  2 \\
  4 \\
  5
\end{array} \right)$} \probpart{Find a basis for the null space of
$A$}\probpart{\textbf{Without carrying out explicit calculation},
does a solution exist for any $b$ in $V^4$? (No credit will be
given for explicit calculation for).}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SPRING 1990}{PRELIM 2}{3}{}
\probb{Find a basis and the dimension of the \underline{column}
space of the matrix $$A = \left[
\begin{array}{ccc}
  1  & 3  & 9   \\
  2  & 6  & 18  \\
  -1 & 1  & 1  \\
  4  & 12 & 36
\end{array} \right]$$} \probpart{Find a basis for the \underline{null} space of the above matrix.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SPRING 1990}{PRELIM 2}{4}{}
\prob{Which of the following sets form a vector subspace of
$V_4$? Explain.} \probpart{the set of vectors of the form
$(x,y,x+y,0)$} \probpart{the set of vectors of the form
$(x,2x,3x,4x)$} \probpart{the set of vectors $(x,y,z,w)$ such
that $x+y+w=1$} \probpart{If the set in $(b)$ is a subspace, find
a basis for it and its dimension.  In the above, $x,y,z,w$ are any
real numbers.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{FALL 1991}{PRELIM 3}{6}{}
\prob{True-False} \probnn{True means always true.  False means
not always true.} \probpart{The column space of a matrix is
preserved under row operations} \probpart{The column rank of a
matrix is preserved under row operations.} \probpart{For an
$n\times n$ matrix, with $m \neq n$, rank plus nullity equals
$n$.} \probpart{The row space of a matrix $A$ is the same vector
space as the row space of the row reduced form of $A$.}
\probpart{If two matrices $A$ and $B$ have the same row space,
the $A=B$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{FALL 1991}{FINAL}{7}{}
\prob{Show that the matrices $A$ and $B$ have the same row space:
\[ A = \left( \begin{array}{ccc}
  3 & 1 & 9 \\
  2 & 1 & 7\\
  1 & 1 & 5
\end{array} \right) , B = \left( \begin{array}{ccc}
  3 & -1 & 3 \\
  1 & -1 & -1\\
  2 & -3 & -5
\end{array} \right) \]}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{FALL 1991}{FINAL}{7}{}
\prob{Find the vector in the subspace spanned by $$ \left\{
\left( \begin{array}{c}
  1 \\
  0 \\
  -1
\end{array} \right) , \left( \begin{array}{c}
  1 \\
  2 \\
  1
\end{array} \right)\right\}$$ which is closest to the vector $$\left( \begin{array}{c}
  2 \\
  -1 \\
  1
\end{array} \right).$$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{FALL 1991}{FINAL}{8}{}
\prob{True-False.  True means always true, false means not always
true.  Warning! Matrices are not necessarily square.}
\probpart{The rank of $A$ equals the rank of $A^T$.}
\probpart{The nullity of $A$ equals the nullity of $A^T$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SUMMER 1992}{FINAL}{3}{}
\prob{Let $V$ be the vector space of all $2 \times 2$ matrices of
the form \[ \left( \begin{array}{cc}
  a_{11} & a_{12} \\
  a_{21} & a_{22}
\end{array} \right) \] where $a_{ij}$,$i,j = 1,2$, are real
scalars.} \probnn{Consider the set $S$ of all $2 \times 2$
matrices of the form \[ \left( \begin{array}{cc}
  a+b & a-b \\
  b & a
\end{array} \right) \] where $a$ and $b$ are real scalars.}
\probpart{Show that $S$ is a subspace. Call it $W$.}
\probpart{Find a basis for $W$ and the dimension of $W$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SUMMER 1992}{FINAL}{3}{}
\prob{Consider the vector space $V$ } \probnn{$ \{ f(t) = a + b
\sin{t} + c \cos{t} \} $,  for all real scalars $a, b$ and $c$
and $0 \leq t \leq 1$} \probnn{Now consider a subspace}
\probpartnn{$W$ of $V$ in which $\frac{d f(t)}{dt} + f(t) = 0$ at
$t=0$} \probnn{Find a basis for the subspace $W$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{FALL 1992}{PRELIM 3}{5}{}
\prob{Fill in the blanks of the following statements.} \probnn{In
what follows $A$ is an $m \times n$ matrix} \probpart{The
dimension of the row space is 2.} \probpartnn{The dimension of
the null space is 3.} \probpartnn{The number of columns of $A$ is
\makebox[0.5in]{\hrulefill} . } \probpart{$Ax = b$ has a solution
$x$ if and only if $b$ is in the \makebox[0.5in]{\hrulefill}
space of $A$. }  \probpart{If $Ax = 0$ and $Ay = 0$ and if $C_1$
and $C_2$ are arbitrary constants then $A(C_1 x + C_2 y)$ =
\makebox[0.5in]{\hrulefill} .}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{FALL 1992}{FINAL}{3}{}
\probb{Let $A$ be an $n \times n$ nonsingular matrix.  Prove that
$\det{\left( A^{-1} \right)} = \frac{1}{\det{(A)}}$.  Hint:  You
may use the fact that if $A$ and $B$ are $n \times n$ matrices
$\det{(AB)} = \det{(A)} \det{(B)}$.} \probpart{An $n \times n$
matrix $A$ has a nontrivial null space.  Find $\det{(A)}$ and
explain your answer.} \probpart{Given two vectors $v_1 = \left[
\begin{array}{c}
  1 \\
  1 \\
  1
\end{array}
\right]$ and $v_2 = \left[
\begin{array}{c}
  1 \\
  0 \\
  1
\end{array}
\right]$ in $V_3$.  Find a vector (or vectors) $w_1, w_2, ...  $
in $V_3$ such that the set $\{v_1, v_2, w_1,...\}$ is a basis for
$V_3$.} \probpart{Let $S$ be the set 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}{SPRING 1993}{PRELIM 3}{6}{}
\prob{Let $A$ be an $n \times n$ matrix. Suppose the rank of $A$
is $r$, and that $\textbf{u}_1,\textbf{u}_2,... ,\textbf{u}_r $
are vectors in $R^n$ such that $A\textbf{u}_1,A\textbf{u}_2,...
,A\textbf{u}_r $ is a basis for $R(A)$ (col. space of A).  Let
$\textbf{v}_1,\textbf{v}_2,... ,\textbf{v}_{n-r} $ be a basis for
$N(A)$ (null space of A).  Then show that $ \{
\textbf{u}_1,\textbf{u}_2,... ,\textbf{u}_r,
\textbf{v}_1,\textbf{v}_2,... ,\textbf{v}_{n-r} \}$ is a basis
for $R^n$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SPRING 1993}{FINAL}{2}{}
\probb{Solve for $y$, for $x$ near $\frac{\pi}{2}$, if
$y^{\prime} + y \cot{x} = \cos{x}$ and $y\left( \frac{\pi}{2}
\right) = 0$} \probpart{Find a basis for the null space of the
differential operator \[ L = \frac{d^2}{dx^2} - 7 \frac{d}{dx} +
12 , -\infty < x < -\infty.
\] (Hint: Find as many linearly independent solutions as needed for the equation $L[y(x)] = 0$.)}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{FALL 1994}{PRELIM 3}{5}{}
\prob{Answer each of the following as True or False. \textbf{If
False, explain, by an example.}} \probpart{Every spanning set of
$R^3$ contains at least three vectors.} \probpart{Every
orthogonal set of vectors in $R^5$ is a basis for $R^5$.}
\probpart{Let $A$ be a 3 by 5 matrix.  Nullity $A$ is at most 3.}
\probpart{Let $W$ be a subspace of $R^4$.  Every basis of $W$
contains at least 4 vectors.} \probpart{In $R^n$, $||cX|| = |c|
||X||$} \probpart{If $A$ is an $n \times n$ symmetric matrix,
then rank $A = n$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{FALL 1994}{FINAL}{4}{}
\prob{A basis for the null space of the matrix $\left[ \begin{array}{ccc}
  1 & 0 & 1 \\
  0 & 1 & 0 \\
  0 & 0 & 0
\end{array} \right]$ is:} \probnn{a. $\left[ \begin{array}{c}
  0 \\
  1 \\
  0
\end{array}\right] b. \left[ \begin{array}{c}
  0 \\
  1 \\
  0
\end{array}\right]\left[ \begin{array}{c}
  1 \\
  0 \\
  1
\end{array}\right] c. \left[ \begin{array}{c}
  1 \\
  0 \\
  1
\end{array}\right] d. \left[ \begin{array}{c}
  1 \\
  0 \\
  -1
\end{array}\right] e. \left[ \begin{array}{c}
  1 \\
  -1 \\
  -1
\end{array}\right]$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{FALL 1994}{FINAL}{8}{}
\prob{If $A$ is an $n$ by $n$ matrix and rank($A$) $<n$. Then}
\probpart{$A$ is non singular,} \probpart{The columns of $A$ are
linearly independent} \probpart{Some eigenvalue of $A$ is zero}
\probpart{$AX=0$ has only the trivial solution} \probpart{$AX=B$
has a solution for every $B$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SPRING 1995}{PRELIM 3}{1}{}
\prob{Consider the matrix \[ A = \left[ \begin{array}{cccc}
  1 & 2 & 4 & 3 \\
  2 & 0 & 4 & -2 \\
  -1 & 3 & 1 & 7
\end{array} \right]. \]} \probpart{Find a basis for the range of $A$ (i.e., the column space of
$A$).} \probpart{Find a basis for the null-space of $A$ (i.e.,
the kernel of $A$).} \probpart{Find a basis for the column space
of $A^T$.}
\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 \[ p(0) + \left. \frac{d p}{d t} \right|_{t=0} =
0.\]} \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}{1}{}
\prob{Consider the matrix \[ A = \left[ \begin{array}{cccc}
  0 & 1 & -1 & 0 \\
  1 & 2 & 0 & 2 \\
  -1 & -1 & -1 & -2
\end{array} \right]. \]} \probpart{Find a basis for the row space of
$A$.} \probpart{Find a basis for the column space of $A$}
\probpart{What is the rank of $A$?} \probpart{What is the
dimension of the null space?}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{FALL 1995}{PRELIM 3}{3}{}
\prob{Let $P_3$ be the space of polynomials $p(t) = a_0 + a_1 t +
a_2 t^2 + a_3 t^3$ of degree $\leq 3$.  Consider the subset $S$
of polynomials that satisfy $$p^{\prime \prime}(0) + 4 p(0) = 0$$
Here $p^{\prime \prime}(0)$ means, as usual, $\left. \frac{d^2
p}{d t^2} \right| _{t=0}$.} \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 constrain, 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 $R^4$ 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 $R^4$.  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 $R^4$. The equations of
$S_1$ and $S_2$ are: } \probpartnn{$S_1: x-u = 0$}
\probpartnn{$S_2: a x + b y + c z + d u = 0$} \probpartnn{where
$\left[ \begin{array}{c}
  x \\
  y \\
  z \\
  u
\end{array} \right]$ is a generic point in $R^4$ 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}{4}{}
\prob{Let \[ A = \left( \begin{array}{cccc}
  1 & 2 & -1 & 3 \\
  2 & 2 & -1 & 2 \\
  1 & 0 & 0  & -1
\end{array} \right) \]} \probpart{Find a basis for the null space of $A$.  What is the dimension of the null space of
$A$?}\probpart{Let$\textbf{x} = (0,\frac{1}{2},1,0)$.  We know
that $A \textbf{x} = \textbf{0}$.  True or false:} \probparts{
$\textbf{x}$ is a trivial solution to $A \textbf{x} =
\textbf{0}.$} \probparts{ $\textbf{x}$ is in the solution space
of $A \textbf{x} = \textbf{0}.$} \probparts{ $\textbf{x}$ is in
the null space of $A$.} \probparts{ \{$\textbf{x}$\} is a basis
for the null space of $A$.} \probpart{The vector \[ \vec{w} =
\left[ \begin{array}{c}
  1 \\
  2 \\
  1
\end{array} \right]
\] is one vector in a basis for the column subspace of $A$. Find another vector $\vec{v}$ in a basis for the column subspace of $A$ such that $\{ \vec{v},\vec{w} \}$ is linearly
independent.} \probpart{What is the rank of $A$? How do you know?}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SPRING 1996}{FINAL}{16}{}
\prob{The vector space of all polynomials of degree six or less
has dimension:} \probpart{5} \probpart{6} \probpart{7}
\probpart{8} \probpart{none of the above}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SPRING 1996}{FINAL}{21}{}
\prob{A basis for the null space of  $\left(
\begin{array}{ccc} 1 & 0 & 1 \\ 0 &1 &0 \\ 0 &0 &0 \end{array} \right)
$ is} \probpart{$\left\{ \left[ \begin{array}{c} 0 \\ 1 \\
0\end{array} \right] \right\}$} \probpart{$\left\{ \left[
\begin{array}{c} 0 \\ 1 \\ 0\end{array} \right], \left[
\begin{array}{c} 1 \\ 0 \\ 0\end{array} \right]\right\}$} \probpart{$\left\{ \left[ \begin{array}{c} 1 \\0\\ -1
\end{array} \right] \right\}$} \probpart{$\left\{ \left[ \begin{array}{c} 1 \\ 0 \\ -1\end{array} \right]
\right\}$} \probpart{none of the above}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SPRING 1996}{FINAL}{23}{}
\prob{Suppose $A$ is a matrix with 6 columns and 4 rows.  Which of
the following must be true?} \probpart{The null space of $A$ has
dimension $\geq 2$ and the rank of $A$ is 4.} \probpart{The null
space of $A$ has dimension $\leq 4$ and the rank of $A$ is 2.}
\probpart{The null space of $A$ has dimension $\leq 2$ and the
rank of $A$ is $\leq 4$.} \probpart{The null space of $A$ has
dimension $\geq 2$ and the rank of $A$ is $\leq 4$.}
\probpart{None of the above}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1997}{FINAL}{2}{}
\prob{(All parts are independent problems)} \probpart{If the
$\det{A}=2$.  Find the $\det{A^{-1}}, \det{A^T}$} \probpart{From
$PA = LU$ find a formula for $A^{-1}$ in terms of $P, L$ and $U$.
Assume $P,L,U,A$ are invertible $n \times n$ matrices.}
\probpart{Find the rank of matrix $A$. \[ A = \left[
\begin{array}{c} 1 \\ 4 \\ 2 \end{array} \right] \left[ \begin{array}{ccc} 2 & -1 & 2\end{array}
\right]\]} \probpart{Find a $2 \times 2$ matrix $E$ such that for
\textit{every} $2 \times 2$ matrix $A$, the second row of $EA$ is
equal to the sum of the first two rows of $A$, e.g. if}
\probpartnn{$A = \left[ \begin{array}{cc} 1 & 2 \\ 3 & 4
\end{array}\right]$ then $EA = \left[ \begin{array}{cc} 1 & 2 \\ 3+1 & 4+2 \end{array}
\right]$} \probpart{Write down a $2 \times 2$ matrix $P$ which
projects every vector onto the $x_2$ axis. Verify that $P^2 = P$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1997}{FINAL}{7}{}
\prob{Suppose $A$ is a 6 row by 7 column matrix for which $nul A
= Span\{\vec{x}_0\} $ for some $\vec{x}_0 \neq \vec{0}$ in
$\Re^7$.  Which of the following are always TRUE of $A$? (NO
Justification is necessary.) Express your answer as e.g: TRUE:
a,b,c,d; FALSE: e} \probpart{The columns of $A$ are linearly
dependent.} \probpart{The linear transformation
$\vec{x}\rightarrow A \vec{x}$ is onto.} \probpart{$A \vec{x} =
\vec{0}$ has only the trivial solution.} \probpart{The columns of
$A$ form a basis for $\Re^6$. } \probpart{The columns of $A$ span
all of $\Re^6$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1997}{PRELIM 2}{1}{}
\prob{Consider the matrix \[ A = \left( \begin{array} {cccc} 4 &
3 & 2 &1\\2&2&0&2\\4&3&1&2\\-2&0&-2&2
\end{array}\right).\]} \probpart{Find a basis for the null space $N$ of $A$.  What is the dimension of
$N$?}\probpart{Find a basis for the column space $C$ of $A$. What
is the dimension of $C$? } \probpart{Find a basis for the row
space $R$ of $A$.  What is the dimension of $R$?}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1997}{FINAL}{2}{}
\prob{Let \[ A = \left( \begin{array}{cccc}
1&2&0&2\\1&1&-1&0\\-2&-1&3&2
\end{array} \right)
\] Find bases for the null space of $A$ and the column space of $A$.  What are the dimensions of these two vector spaces?}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1998}{PRELIM 3}{1}{}
\prob{The matrix $A$ is row equivalent to the matrix $B$: \[ A
\equiv \left[ \begin{array}{ccccc}
1&0&-5&1&4\\-2&1&6&-2&-2\\0&2&-8&1&9
\end{array}\right] \sim \left[ \begin{array}{ccccc}
1&0&-5&1&4\\0&1&-4&0&6\\0&0&0&1&-3
\end{array}\right]\equiv B
\]} \probpart{Find a basis for $Row A$, $Col A$, and $Nul A$.}
\probpart{To what vector spaces do the vectors in $Row A$, $Col
A$, and $Nul A$ belong?} \probpart{What is the rank of $A$?}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1998}{FINAL}{3}{}
\prob{Given that the matrix $B$ is row equivalent to the matrix
$A$ where} \probpartnn{$A \equiv \left[ \begin{array}{ccccc}
2&-1&1&-6&8\\1&-2&-4&3&-2\\-7&8&10&3&-10\\4&-5&-7&0&4
\end{array}\right]$ and $B \equiv \left[ \begin{array}{ccccc}
1&-2&-4&3&-2\\0&3&9&-12&12\\0&0&0&0&0\\0&0&0&0&0
\end{array}\right]$} \probpart{Find rank $A$ and dim Null $A$.}
\probpart{Determine bases for Col $A$ and Null $A$.}
\probpart{Determine a value of $c$ so that the vector
$\vec{b}=\left[ \begin{array}{c} 1\\-1\\1\\c \end{array}\right]$
is in Col $A$} \probpart{For this value of $c$, write the general
solution of $A \vec{x} = \vec{b}$. }
\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 3}{1}{}
\prob{Let \[ A = \left( \begin{array}{cccc} 1&1&-1&1\\2&1&2&1
\end{array} \right).\]} \probpart{Find an orthofonal basis for the null space of
$A$.}\probpart{Find a basis for the orthogonal complement of $Nul
A $, i.e. find $(Nul A)^\perp.$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1997}{PRELIM 3}{2}{}
\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 \[ A^T A =
\left( \begin{array}{cc} 1& \frac{1}{2} \\ \frac{1}{2}&1
\end{array} \right).
\] Find an orthonormal basis $\{ \vec{u}_1, \vec{u}_2\}$ of the column space of $A$.  Your formulas for $\vec{u}_1$ and $\vec{u}_2$ should be written as linear combinations of $\vec{v}_1, \vec{v}_2$.  (Hint: what do the entries of the matrix $A^T A$ have to do with dot products?}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1998}{PRELIM 2}{4}{}
\prob{The reduced echelon form of the matrix $A = \left[
\begin{array}{cccc} 3&3&2&3\\-2&2&0&2\\1&0&1&-2\\0&-3&2&-1 \end{array} \right]$ is $B = \left[ \begin{array}{cccc} 1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1 \end{array}
\right].$}\probpart{What is the rank of $A$} \probpart{What is
the dimension of the column space of $A$?} \probpart{What is the
dimension of the null space of $A$?} \probpart{Find a solution to
$A \vec{x} = \left[ \begin{array}{c} 3\\-2\\1\\0 \end{array}
\right].$ } \probpart{What is the row space of $A$?}
\probpart{Would any of your answers above change if you changed
$A$ by randomly changing 3 of its entries in the 2nd, third, and
fourth columns to different small integers and the corresponding
reduced echelon form for $B$ was presented? (yes?,  no?,
probably?, probably not?, ?)}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALLL 1998}{FINAL}{5}{}
\prob{Consider $A \vec{x} = \vec{b}$ with $A = \left[
\begin{array}{cccc} 1&2&3&4\\0&1&2&3\\-1&2&5&8 \end{array}\right]$
and $\vec{b} = \left[ \begin{array}{c} 0\\1\\1 \end{array}
\right]$. The augmented matrix of this system is $\left[
\begin{array}{ccccc} 1&2&3&4&1\\0&1&2&3&1\\-1&2&5&8&1
\end{array} \right]$ which is row equivalent to $\left[
\begin{array}{ccccc} 1&0&-1&-2&0\\0&1&2&3&0\\0&0&0&0&1
\end{array} \right]$.} \probpart{What are the rank of $A$ and dim
nul $A$? (Justify your answer.)} \probpart{Find bases for col
$A$, row $A$, and nul $A$.} \probpart{What is the general
solution $\vec{x}$ to $A \vec{x} = \vec{b}$ with the given $A$
and $\vec{b}$?} \probpart{Select another $\vec{b}$ for which the
above system has a solution. Give the general solution for that
$\vec{b}$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1999}{PRELIM 2}{4}{}
\prob{Let $A$ be a matrix where all you know is that it is 5
$\times$ 7 and has rank 3.} \probpart{Define new matrices from
$A$ as follows:
\begin{itemize}
  \item $C$ has as columns a basis for Col $A$,
  \item $M$ has as columns a basis for Nul $A^T$, and
  \item $T = [CM].$
\end{itemize}
Is this enough information to find the size (number of rows and
columns) of $T$?} \probparts{if yes, find the number of rows and
columns and justify your answer, or} \probparts{if no, explain
what extra information is needed to find the size of $T$?}
\probpart{Are there any two non-zero vectors $\vec{u}$ and
$\vec{v}$ for which:
\begin{itemize}
  \item $\vec{u}$ is in Col $A$,
  \item $\vec{v}$ is in Nul $A^T$, \textit{\textbf{and}}
  \item $\vec{v}$ is a multiple of $\vec{u}$?
\end{itemize}
} \probparts{if yes, why?} \probparts{if no, why?, or}
\probpart{if it depends on information not given, what
information? How would that information help?}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1999}{PRELIM 2}{1}{}
\probb{What is the null space of $A = \left[ \begin{array}{ccc}
0&0&0\\0&0&0\\0&1&0\end{array} \right]$?} \probpart{What is the
column space of $A = \left[ \begin{array}{ccc}
1&0&0\\0&0&0\\0&0&1\end{array} \right]$} \probpart{Find a basis
for the column space of $A = \left[ \begin{array}{ccc}
1&2&\pi\\3&4&\sqrt{2}\end{array} \right]$?} \probpart{Are the
column of $A = \left[
\begin{array}{ccc}1&2&\pi\\3&4&\sqrt{2}\\3&4&\sqrt{2}
\end{array}\right]$ linearly independent (hint: no long row reductions are
needed)?}\probpart{What is the row space of $A = \left[
\begin{array}{cc} 1&0\\4&0 \end{array}\right]$?}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SUMMER 1992}{PRELIM 7/21}{5}{}
\prob{Consider the space $P$ of all polynomials of degree $\leq
3$ of the type $\{ p(t) = a_0 + a_1 t + a_2 t^2 + a_3 t^3 \}$ for
all scalars $a_0, a_1, a_2, a_3$ and $0 \leq t \leq 1$. Now
consider a subspace $W$ of $P$ where, for any $p(t) \in W,$ we
also have \[ \int^1_0 p(t) d t = 0 \] \[ \left.\frac{d p}{d
t}\right|_{t=0} = 0
\]} \probpart{Find a basis for $W$.} \probpart{What is the dimension of $W$?}
\end{problem}
%----------------------------------
\begin{problem}{UNKNOWN}{UNKNOWN}{UNKNOWN}{?}{}
\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 $A \vec{x} = \vb$
exists.}\probpart{Find a basis for the column space
$\mathcal{R}(A)$ of $A$.} \probpart{It is claimed that
$\mathcal{R}(A)$ is a plane on $\Re^3$. If you agree, find a
vector $\vec{n}$ in $\Re^3$ that is normal to this plane. Check
your answer.} \probpart{Show that $\vec{n}$ is perpendicular to
each of the columns of $A$. Explain carefully why this is true.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{97}}{\pr{3}}{1 PRACTICE}{}
\prob{Consider the matrix \[ A = \sqbrc{\begin{array}{cccc}
   0 &  1 & -1 &  0 \\
   1 &  2 &  0 &  2 \\
  -1 & -1 & -1 & -2
\end{array}}\]} \probpart{Find a basis for the column space C of A. What is the dimension of
C?} \probpart{Find a basis for the column space N of A. What is
the dimension of N?} \probpart{Let $W =
span\brc{\vtfour{1}{0}{2}{2}, \vtfour{0}{1}{-1}{0}}$. Is $W$
orthogonal to N? Please justify your answer by showing your work.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SPRING ?}{\pr{2}}{1}{}
\probb{Find a basis for the row space of the matrix \[A =\
\sqbrc{\begin{array}{rrrr}
  1 & -1 & -1 & 1 \\
  0 &  1 &  2 & 1 \\
  3 &  2 &  7 & 8 \\
  2 &  0 &  2 & 4
\end{array}}
\]} \probpart{Find a basis for the column space of A in (a).}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SPRING ?}{\pr{2}}{2}{}
\probb{If $A \and B$ are $4 \times 4$ matrices such that \[ A B
=\brc{\begin{array}{rrrr}
   2 &  1 &  1 & 0 \\
  -1 & -2 &  2 & 0 \\
   0 &  0 &  3 & 0 \\
   0 &  0 &  0 & 0
\end{array}},
\] show that the column space of $A$ is at least three dimensional.} \probpart{Find $A^{-1}$ if $A = \brc{\begin{array}{rrrr}
   2 & -1 &  0 & 0 \\
  -1 &  2 & -1 & 0 \\
   0 & -1 &  2 & -1 \\
   0 &  0 & -1 & 1
\end{array}}$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SPRING ?}{FINAL}{2}{}
\probb{Find a basis for the row space of the matrix \[ A =
\brc{\begin{array}{rrrr}
   1 &  1 &  0 & 1 \\
   1 &  0 &  0 & 2 \\
   0 &  0 &  4 & 0 \\
   1 &  2 &  0 & 0
\end{array}} \]} \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}
%----------------------------------
\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 $(\vec{u},
\vv)$ of vectors, for $\vu, \vv$ 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}{SPRING ?}{FINAL}{3}{}
\probb{Give all solutions of the following system in vector form.
\[ \begin{array}{rcl}
  6 x_1 + 4 x_3 & = & 1\\
  5 x_1 - x_2 + 5 x_3 & = & -1 \\
  x_1 + 3 x_3 & = & 2
\end{array} \] } \probpart{What is the null space of the matrix of coefficients of the unknowns in a)?}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SPRING ?}{FINAL}{4}{}
\prob{Let $W$ be the subspace of $V_4$ spanned by the vectors \[
\vv_1 =\vtfour{1}{0}{-1}{2} , \; \vv_2 = \vtfour{-3}{0}{1}{1}, \;
\vv_3 = \vtfour{7}{0}{-3}{3}, \; \vv_4 = \vtfour{-8}{0}{4}{1}
\]} \probpart{Find the dimension and a basis for $W$.} \probpart{Find an orthogonal basis for $W$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{UNKNOWN}{FINAL}{5}{}
\probb{Let $A$ be an $n \times n$ matrix. Show that if $A \vec{x}
= \vb $ has a solution then $\vb$ is a linear combination of the
column vectors of $A$.} \probpart{Let $A$ be a $4 \times 4$
matrix whose column space is the span of vectors $\vv =(v_1, v_2,
v_3, v_4)^T$, satisfying $v_1 - 2 v_2 + v_3 - v_4 = 0.$ Let $\vb =
(1, b_2, b_3, 0)^T.$ Find all values of $b_2, b_3$ for which the
matrix equation $A \vec{x} = \vec{b}$ has a solution.}
\end{problem}
