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\begin{document}
Sept. 5, 1995 \hfill Math 311, Quiz 1 \hfill
\sk
\q{5}Find a cubic polynomial whose graph passes through the following
points: $(-2,-8)$, $(-0.5,4)$, $(0.5,7)$, and $(3,-3)$. Can you find a second
degree polynomial whose graph passes through these same points?
\sk
\q{5}Let
$A=\left[\begin{array}{rrrr}1&2&1&0\\2&-1&3&5\\1&1&1&1\end{array}\right]$.
Find a sequence of elementary row matrices, $E_1,\cdots,\ E_k$, such
that $E_1E_2\cdots E_kA$ is in row echelon form. What is the inverse
of the matrix $E_1E_2\cdots E_k$?
\num=1
\Sk
Sept. 12, 1995 \hfill Math 311, Quiz 2 \hfill
\sk
\q{5}Let
$A=\left[\begin{array}{rrr}2&-1&3\\4&-2&-3\\4&4&4\end{array}\right]$.
Does $A$ have an LU factorization? Note, L represents a lower
triangular matrix with ones on the main diagonal. If yes, find it. If
not, does there exist a permutation matrix $P$ such that $PA$ has an LU
factorization. Again, L denotes a lower triangular matrix with ones on
the main diagonal. 
\sk
\q{5}Suppose the combined population of Bryan-College Station is always
100,000. Suppose that each year 35\% of the Bryan residents move to
College Station and 40\% of the College Station residents move to
Bryan. Assuming that both cities start with the same number of
residents, what will the populations of each city be after 5 years,
after 25 years?
\num=1
\Sk
Sept. 19, 1995 \hfill Math 311, Quiz 3 \hfill 
\sk
\q{2}Find the projection of $[1,2,3,4]$ onto $[-2,0,3,7]$. Use the
standard inner product.
\sk
\q{8}Define the following inner product on $R^3$:
$<\vec{x},\vec{y}>=x_1y_1-x_2y_1+2x_1y_2 +x_2y_2 + x_3y_3$.
\begin{list}{\alph{bean}. }{\usecounter{bean}}
\item Show that this is an inner product. {\bf Messed up, this form is
not symmetric}.
\item Compute the norm of the vector $[1,2,3]$
\item Compute the projection of the vector $[3,-4,1]$ onto $[1,2,3]$.
\end{list}
\num=1
\newpage
Sept. 21, 1995 \hfill Math 311, Quiz 4 \hfill 
\sk
\q{4}Let $W$ be that subset of $P_4$ which consists of all polynomials
that are zero at $t=0$ and $t=2$.
\begin{list}{\alph{bean}. }{\usecounter{bean}}
\item Show that $W$ is a subspace of $P_4$.
\item Find a set of vectors that span $W$. Try to make your set
contain as few vectors as possible.
\end{list}
\sk
\q{6}Define the following ``inner product'' on $P_3$: $\ds
<p,q>=\int_0^2p(t)q(t)\,dt$.
\begin{list}{\alph{bean}. }{\usecounter{bean}}
\item Show that this is an inner product.
\item Let $p(t) = 2t-3$, and $q(t) = t^2-3t + 2$. Find the length of
the vector $q$.
\item Find the projection of $p$ onto $q$.
\end{list}
\num=1
\Sk
Sept. 26, 1995 \hfill Math 311, Quiz 5
\sk
\q{6}Let $\ds W=\left\{\,p\in P_3\,:\,\mbox{such that
}\int^2_0p(t)\,dt =0\,\mbox{ and }p^\prime(1)=0\,\right\}$. 
\begin{list}{\alph{bean}. }{\usecounter{bean}}
\item Show that $W$ is a subspace of $P_3$, and then determine the
dimension of $W$.
\item Find an orthonormal basis for $W$. Note: $\ds
<p,q>=\int_0^2p(t)q(t)\,dt$. 
\item Extend your orthonormal basis of $W$ to an orthonormal basis of
$P_3$.
\end{list}
\q{4}Let $W$ be that subset of $M_{3,3}$ such that if $A\in W$, then
the trace of $A$ and the trace of $A^T$ both equal zero.
\begin{list}{\alph{bean}. }{\usecounter{bean}}
\item Show that $W$ is a subspace of $M_{3,3}$ and find a basis for $W$.
\item The matrix
$A=\left[\begin{array}{rrr}1&2&-3\\4&6&0\\-3&1&-7\end{array}\right]$ is in
$W$. Find its coordinates with respect to the basis you find in part a.
\end{list}
\num=1
\Sk
Oct, 3, 1995 \hfill Math 311, Quiz 6 \hfill 
\sk
\q{6}Let $A=\left[\begin{array}{rrrr}6&1&2&4
\\-3&0&1&8\end{array}\right]$. Use the standard inner product on the
vector space $R^4$.
\begin{list}{\alph{bean}. }{\usecounter{bean}}
\item Find an orthogonal basis for the row space of $A$.
\item Find an orthogonal basis for the null space of $A$.
\end{list}
\sk
\q{4}Let $A=\left[\begin{array}{rrr}4&0&3
\\0&2&-1\\3&-1&5\end{array}\right]$. Define the following inner
product on $R^3$:
$<\vec{x},\vec{y}>=\left(A\vec{x}\right)\cdot\left(\vec{y}\right)$.
Where the ``dot'' on the right side of the equals sign signifies the
standard inner product on $R^3$. You might convince yourself that this
is indeed an inner product.
\begin{list}{\alph{bean}. }{\usecounter{bean}}
\item What is the length of the vector $(1,0,0)$? Are the two vectors
$(1,0,0)$ and $(0,1,0)$ orthogonal?
\item From the standard basis of $R^3$ construct an orthogonal basis
of $R^3$.
\end{list}
\num=1\sk
Oct, 19, 1995 \hfill Math 311, Quiz 7 \hfill 
]
\sk
\q{5}Find the matrix representation with respect to the standard basis
of $R^3$ of the linear transformation, $L$, which does the following:
rotates the plane $W=\{(x_1,x_2,x_3):x_1+x_2+x_3=0\}$ through an angle
of 30 degrees in a counterclockwise direction when viewed from the
point $(1,1,1)$. At the same time $L$ stretches $R^3$ by a factor of 3
in the direction of the vector $(1,1,1)$. Hint: find an orthonormal
basis of $W$.
\sk
\q{5}Let $L$ be a linear transformation of $P_2$ into $R^4$ such that
$L(\vec{p})=(p(1),p(2),p^\prime(1),\int_0^2p(t)\,dt)$. Find the matrix
representation of $L$ with respect to the standard bases of $P_2$ and
$R^4$.  
\num=1
\Sk
Oct, 26, 1995 \hfill Math 311, Quiz 8 \hfill
\sk
\q{5}Let $A=\left[\begin{array}{rrrr}.25&.75&.75&-.75\\0&1.5&-.5&-1\\
0&0&1&0\\0&.5&-.5&0
\end{array}\right]$.
\sk
\begin{list}{\alph{bean}. }{\usecounter{bean}}
\item $\ds \lim_{n\to\infty}A^n =$?
\item $e^A =$?
\end{list}
\sk
\q{5}Define the following inner product on $P_3$ by $\ds
<p,q>=\sum_{i=-1}^2p(i)q(i)$. Let $B = \{1,t,t^2\}$.
\sk
\begin{list}{\alph{bean}. }{\usecounter{bean}}
\item Find the matrix $A$ which satisfies: $<p,q>=[p]_B\cdot A[q]_B$,
where $[p]_B$ denotes the coordinates of $p$ with respect to the basis $B$.
\item Verify that $A$ is a positive definite matrix. Note: I do not
want you saying, ``Well, $A$ comes from an inner product and from our
class notes we know that $A$ is positive definite.''. Verify directly that
your $A$ is positive definite.
\item Find an orthogonal basis of $P_3$.
\end{list}
\num=1
\Sk
Nov. 2, 1995 \hfill Math 311, Quiz 9 \hfill 
\sk
\q{6}Let $\Gamma(t)=(x(t),y(t),z(t))$ describe the path of a
bumblebee. Suppose that $\Gamma(0)=(1,-1,2)$,
$\Gamma^\prime(0)=\vec{0}$, and that the bee's
acceleration is given by the formula:
$$
\Gamma^{\prime\prime}(t)=\left\{\begin{array}{rl}\ds\frac{t}{4}(1,0.25,0.1),&0\leq
 t\leq 1\\ {}\\(0,0,0),& 1<t
\end{array}\right.\,.
$$
\begin{list}{\alph{bean}. }{\usecounter{bean}}
\item Where is the bee located and what is its velocity when $t=3$?
\item Find the unit tangent vector to the path of the bee.
\item Find the unit normal to the path of the bee.
\end{list}
\sk
\q{4}Let $f(x_1,x_2)=x_1^2 - 3x_1x_2 + x_2^4$.
\begin{list}{\alph{bean}. }{\usecounter{bean}}
\item Find all values of $x_1,\mbox{ and }x_2$ for which $\del
f=\vec{0}$, that is, the gradient
of $f$ equals the zero vector.
\item Suppose we have an ant sitting at the point $(1,1)$. What is the
path the ant will follow if its velocity vector at any point
$(x_1,x_2)$ equals $\del f$. Determine where the ant is after one time
interval. 
\end{list}
\num=1
\Sk
Nov. 9, 1995 \hfill Math 311, Quiz 10 \hfill 
\sk
\q{6}Let $f(x_1,x_2,x_3)=x_1^2+3x_2^2+5x_3^2+4x_1-18x_2-30x_3$, and
let $g(x_1,x_2,x_3)=2x_1^2+3x_2^2+x_3^2$. Locate all relative and
absolute extremum of the function $f$ on the set $g\leq 100$.
\sk
\q{4}Let $\phi(x,y)=x^2y-e^x\cos y$. Find as many points $(x,y)$ as
you can at which $\del\phi=\vec{0}$.
\num=1
\Sk
Nov. 16, 1995 \hfill Math 311, Quiz 11 \hfill
\sk
Let $\ds T(u,v)=\left(\frac{u^2+v^2}{2},\,uv\right)=(x,y)$.
\sk
\q{2}Show that the mapping $T$ maps the region $\Omega_U=\{(u,v):0\leq
u+v\mbox{ and }0\leq u-v\,\}$ in a one-to-one manner onto the region
$\Omega_X=\{(x,y):0\leq x+y\mbox{ and }0\leq x-y\,\}$.
\sk
\q{2}Solve the equation $T(u,v)=(2,1)$. That is find values $u_0$ and
$v_0$ such that $T(u_o,v_0)=(2,1)$, and then graph the
curves $T(u_0,v)$, and  $T(u,v_0)$ which lie in the set
$\Omega_X$. We are only interested in values of $u$ and $v$ which lie
close to $u_0$ and $v_0$. 
\sk
\q{3}Find unit vectors $\vec{e}_u$ and $\vec{e}_v$ which point in the
direction of increasing $u$ and $v$ at the point $(2,1)$.
\sk
\q{3}Find a formula for computing the gradient of a function in terms
of the $u$-$v$ coordinate system at an arbitrary point in $\Omega_X$.
\num=1
\Sk
Nov. 21, 1995 \hfill Math 311, Quiz 12 \hfill 
\sk
\q{5}Let $V=\{\vec{x}\in R^4\,:\,x_1+x_2+x_3+x_4=0\mbox{ and
}x_1-x_2-5x_3+3x_4=0\,\}$. Let $L$ be the linear transformation from
$R^4$ to $R^4$ such that $L(\vec{x})=\mbox{ Proj}_V\vec{x}$. That is,
$L$ is the projection onto the subspace $V$.
\begin{list}{\alph{bean}. }{\usecounter{bean}}
\item Find the matrix representation of $L$ with respect to the
standard basis of $R^4$.
\item What is $L((1,2,3,4))$?
\end{list}
\sk
\q{5}Let $f(x,y,z)=3x^2+y^2-18x+15y+3z^2+15z-7$. Let $\Omega =
\left\{\,(x,y)\,:\,|x|\leq 10,\ |y|\leq 12,\mbox{ and }|z|\leq5\,\right\}$.
\begin{list}{\alph{bean}. }{\usecounter{bean}}
\item Find the absolute maximum and minimum values of $f$ on all of
$R^3$, if they exist, and their locations. Find the locations of all
relative extremum of the function $f$.
\item Find the locations and values of the the largest and smallest
values of $f$ on the set $\Omega$. Hint: each of the sides of $\Omega$
can be described as the level surface of some function.
\end{list}
\num=1
\Sk
Nov. 30, 1995 \hfill Math 311, Quiz 13 \hfill 
\sk
Let $S$ be the surface that is that part of the graph of the function
$f(x,y)=x^2+2xy-2y+6$ which lies above the region 
$\Omega=\{(x,y) : 0\leq x\leq 2 \mbox{ and }1\leq y\leq 2\}$.
\sk
\q{2}Find the area of the surface $S$.
\sk
\q{6}Suppose this surface represents a piece of deer pelt, and that
the mass density function of the pelt equals
$\rho(x,y,z)=1+x+y-.05z^2$. What are the coordinates of the center of
mass of the pelt?
\sk
\q{2}What is the maximum value of the mass density function of the deer
pelt in the preceding problem?



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