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\begin{document}
January 24, 1995 \hfill Math 311, Quiz 1\sk
\q{5}Is it possible to pass a fourth degree polynomial through the
following set of points:
$$\{[1,1],[-1,1],[0,3],[-2,5],[3,5],[4,4],[5,6]\}?$$
 If yes, find such a
polynomial, and if no, why not?
\sk
\q{3}Find, if possible, a nonzero vector in $R^4$ that is
perpendicular to each of the 
following vectors: $\{[1,1,1,1],[1,0,2,1],[0,1,2,3]\}$.
\sk
\q{2}Find, if possible, a nonzero vector in $R^4$ that is
perpendicular to each of the 
following vectors: $\{[1,1,1,1],[1,0,2,1],[0,1,2,3],[1,0,0,0]\}$.
\num=1
\Sk
January 31, 1995 \hfill Math 311, Quiz 2 \hfill
\sk
\q{5}Show that the mapping from $R^7$ into ${\cal C}[R]$ given by:
$$
L(x_1,x_2,x_3,x_4,x_5,x_6,x_7)= x_1+x_2\sin(x)+x_3\sin(2x)+x_4\sin(3x)
+x_5\cos(x)+x_6\cos(2x)+x_7\cos(3x)$$
is a linear function.
\sk
\q{5}Is there a function $f(x)$ in the image of $L$ whose graph passes
through the following points:
$$
\{\,(1,1),\ (2,1),\ (3,5),\ (4,6),\ (5,0),\ (6,-4),\ (7,3)\,\}\,?
$$
\num=1
\Sk
February 7, 1995 \hfill Math 311, Quiz 3 \hfill 
\sk
Let $A=\left[
\begin{array}{rrr}2&3&-1\\-5&2&1\\2&0&13\\-3&7&0\end{array}\right]$.
\sk
\q{2}Is the vector $[1,2,0,3]$ in the span of the columns of $A$?
\sk
\q{4}Find a finite set of vectors whose span is the null space of $A$.
\sk
\q{4}Find a finite set of vectors whose span is the image of $A$.
\num=1
\newpage
February 14, 1995 \hfill Math 311, Quiz 4 
\sk
\q{6}Find a basis for each of the following vector spaces:
\newcounter{bean}
\begin{list}{\alph{bean}. }{\usecounter{bean}}
\item The null space of the matrix $A=\left[
\begin{array}{rrrr}3&1&0&2\\-1&1&-2&1\\2&0&-3&5
\end{array}\right]$.
\item The image space of the matrix: $A=\left[
\begin{array}{rrrr}3&1&0&2\\-1&1&-2&1\\2&0&-3&5
\end{array}\right]$.
\item The vector space V which consists of all two by three matrices
whose 2,2 and 1,3 entries equal zero.
\end{list}
\sk
\q{4}Define $L:P_2\rightarrow R^3$ by $L(\vec{p}) =
(p(-2),p(1),p(2))$.
\begin{list}{\alph{bean}. }{\usecounter{bean}}
\item Show that $L$ is a linear transformation.
\item Find the matrix representation of $L$ with respect to the
standard bases of $P_2$ and $R^3$. These are $\{\,1,\ t,\ t^2\,\}$ and
$\{\,\vec{e}_1,\ \vec{e}_2,\ \vec{e}_3\,\}$ respectively.
\end{list}
\num=1
\Sk
February 16, 1995 \hfill Math 311, Quiz 5 \hfill 
\sk
\q{10}Find a basis for each of the following vector spaces:
\begin{list}{\alph{bean}. }{\usecounter{bean}}
\item The null space of the matrix $A=\left[
\begin{array}{rrrr}3&1&0&2\\-1&1&-2&1\\2&0&-3&5
\end{array}\right]$.
\item The image space of the matrix: $A=\left[
\begin{array}{rrrr}3&1&0&2\\-1&1&-2&1\\2&0&-3&5
\end{array}\right]$.
\item The vector space V which consists of all two by three matrices
whose 2,2 and 1,3 entries equal zero.
\item Let $B=\left[\begin{array}{cc}1&-1 \\2&3\end{array}\right]$.
That subspace of $M_{2,3}$ which consists of all matrices $A$ such
that $BA=O$.
\item The subspace of $P_4$ which consists of those polynomials $p(t)$
such that $p(-1)=0,\ p(1)=0,\ p(3)=0$.
\end{list}
\num=1\Sk
February 23, 1995 \hfill Math 311, Quiz 6 \hfill
\sk
Let $B_1 = \{\,t^2+1,\ t+3,\ t^3 + 1,\ 2t\,\}$ and 
$B_2 = \{\,1,\ t,\ t^2,\ t^3\,\}$.
\sk
\q{2}Show that the sets $B_1$ and $B_2$ are bases of $P_3$, the vector
space of polynomials of degree three or less.
\sk
\q{2}Find the change of basis matrix $Q$ such that $[\vec{x}]_{B_1} =
Q[\vec{x}]_{B_2}$. 
\sk
\q{3} Define the linear transformation $L:P_3\rightarrow P_3$ by:
$$
L(\vec{p})=D(p) + \frac1x\int_0^x p(t)\,dt\,;
$$
where the symbol $D$ denotes differentiation. Thus, if $p(x)=x$, then
$\ds L(\vec{p})= 1 + \frac{x}{2}$.
Show that $L$ is a linear transformation from $P_3$ into $P_3$.
\sk
\q{3}Find the matrix representation of $L$ with respect to the basis $B_1$.
\num=1
\Sk
March 23, 1995 \hfill Math 311, Quiz 7 \hfill 
\sk
\q{10}An ant is walking across your desk and its position at time t is
given by the position function 
$$
P(t) =
\left(\frac{t^4}{4}-2t^3+\frac{11}{2}t^2-6t+1.5,1-\cos(t^2)\right)\,,$$
where t is in seconds and one unit of length is one inch. Assume the
edge of the desk nearest you is the x--axis and the y axis is the middle
line of your desk.
\begin{list}{\alph{bean}. }{\usecounter{bean}}
\item Clearly the farthest the ant gets from the edge 
nearest you is 2 inches. How many inches has the ant moved before it
gets two inches from the edge the first time?
\item When it crosses from the left half plane to the right half plane
the first time, what is its velocity, and how long has it been walking
before this happens?
\item At what times and locations is the velocity of the ant equal to zero?
\end{list}
\num=1
\Sk
April 4, 1995 \hfill Math 311, Quiz 8 \hfill 
\sk
\q{5}Let $F(x,y,z,w) = 2x+3y-4z+w^2$. Find all extremum of $F$
subject to the constraint, $g(x,y,z,w) = x^2+y^2+z^2+w^2 \leq 4$.
\sk
\q{5}Let $f(x,y) = x^2+2xy-\sin(y)$. Compute the gradient of $f$ in
polar coordinates at the point $(1,2)$.
\num=1
\newpage
April 13, 1995 \hfill Math 311, Quiz 9 \hfill 
\sk
\q{5}Let $f(x,y,z)=2x^2+3|y| + z^2$ be the mass density function of a
region $R$ in $R^3$, where $R= \{\,(x,y,z):|x|+|y|\leq 1,\ 0\leq z \leq
1\,\}$.
\begin{list}{\alph{bean}. }{\usecounter{bean}}
\item Sketch the region $R$ being particular about that part of $R$ in
the $z=0$ plane.
\item What is the mass of the solid to three decimal places?
\item To three decimal places, what is the volumn of the region?
\item To three decimal places, what is the z coordinate of the center
of mass of the solid? 
\end{list}
\sk
\q{5} Two particles move from the origin in $R^3$ to the point with
coordinates $(1,1,1)$. The first particle moves along the straight
line path joining the two points. The second particle travels the
straight line paths in the following manner: $(0,0,0)\rightarrow
(1,0,0)\rightarrow (1,0,1)\rightarrow(1,1,1)$. While they are
traveling both particles are subjected to the force field
$F(x,y,z)=(xy,x^2+y^2,yz)$. Find the work done by the force field on
each of the two particles.
\num=1
\Sk
April 25, 1995 \hfill Math 311, Quiz 10 \hfill 
\sk
Let $F(x,y,z)= (x^2y+z,2x-3zy,z^2)$. 
\sk
\q{3}Find the work done by $F$ on a particle that travels once about
the circle in the $x,y$ plane centered at the origin of radius 2.
\sk
\q{2}Find the integral of the divergence of $F$ over the interior of
the circle given in the previous problem.
\sk
\q{3}Find the work done by $F$ on a particle that travels once about
the circle centered at the origin of radius 2. This time the circle
lies in the plane whose equation is $x+y+z=0$.
\sk
\q{2}Find the integral of the divergence of $F$ over the interior of
the circle given in the previous problem.
\num=1
\Sk
April 27, 1995 \hfill Math 311, Quiz 11 \hfill 
\sk
Let $F(x,y,z)= (x^2y+z,2x-3zy,z^2)$. 
\sk
\q{3}Find the work done by $F$ on a particle that travels once about
the circle in the $x,y$ plane centered at the origin of radius 2.
\sk
\q{2}Find the integral of the normal component of the curl of $F$ over
the interior of the circle given in the previous problem.
\sk
\q{3}Find the work done by $F$ on a particle that travels once about
the circle centered at the origin of radius 2. This time the circle
lies in the plane whose equation is $x+y+z=0$.
\sk
\q{2}Find the integral of the normal component of the curl of $F$ over
the interior of the circle given in the previous problem.
\end{document}