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\title[Problem Sets 15--17]{Problem Sets 15 to 17.  Due as specified.}

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\centerline{\sc Problem Set 15: Due 26 September 2000.}

\noindent {\bf Supplementary reading.} Strang, last two sections of
Chapter 3.

\noindent {\bf Reading for PS 16.} Strang, Chapter 4 and section 7.3.

\noindent {\bf Reading for PS17.} {\em Matrices and Transformations,}
sections 4--1 to 4--4.  Strang, sections 6.1 and 6.2.

\begin{enumerate}

\item Which of the following sets of vectors span $\R^3$?
\begin{enumerate}
\item $(1,2,0$ and $(0,-1,1)$.
\item $(1,1,0)$, $(0,1,-2)$, and $(1,3,1)$.
\item $(-1,2,3)$, $(2,1,-1)$, and $(4,7,3)$.
\item $(1,0,2)$, $(0,1,0)$, $(-1,3,0)$, and $(1,-4,1)$.
\end{enumerate}

\item Which of the following sets of vectors span $P_3=\{
at^3+bt^2+ct+d\}$?
\begin{enumerate}
\item $t+1$, $t^2-t$, and $t^3$.
\item $t^3+t$ and $t^2+1$.
\item $t^2+t+1$, $t+1$, $1$, and $t^3$.
\item $t^3+t^2$, $t^2-t$, $2t+4$, and $t^3+2t^2+t+4$.
\end{enumerate}

\item Are the following sets of vectors linearly dependent or
independent?  If they are dependent, write one as a linear combination of
the others.
\begin{enumerate}
\item $(1,2,0)$ and $(0,-1,1)$ in $\R^3$.
\item $(-1,2,3)$, $(2,1,-1)$, and $(4,7,3)$ in $\R^3$.
\item $(1,2)$, $(2,3)$, and $(8,-2)$ in $\R^2$.
\item $t^2+2t+1$, $t^3-t^2$, $t^3+1$, and $t^3+t+1$ in $P_3$.
\end{enumerate}

\item What is the dimension of the following spaces?
\begin{enumerate}
\item The set of $2\times 2$ symmetric matrices, $A=A^T$.
\item The set of $2\times 2$ matrices
$$
A=\left[\begin{array}{cc}a&b\\c&d\end{array}\right],
$$
with $a+d=0$.
\item The set $\{ (x,y,x-3y,2y-x)\ |\ x,y\in\R\}$ inside of $\R^4$.
\end{enumerate}

\item What is the column space and row space of the matrix
$$
A=\left[\begin{array}{rrrr}1&3&5&-2\\2&-1&3&-4\\-1&4&2&2\end{array}\right]?
$$

\item Find an (infinite) basis for the space of all polynomials
$$
\mathcal{P}=\{ a_nx^n+z_{n-1}x^{n-1}+\cdots+a_1x+a_0\ |\ \mbox{ for all }
n\}.
$$

\item Suppose $\{ v_1,\dots,v_n\}$ spans a vector space $V$, and suppose
that $v_n$ is a linear combination of $v_1$ through $v_{n-1}$.  Then show
that $\{ v_1,\dots,v_{n-1}\}$ spans $V$ as well.

\item If $A$ is a $4\times 6$ matrix, show that the columns of $A$ are
linearly dependent.

\item Compute $$
\left[\begin{array}{rr}.1&.95\\.9&.05\end{array}\right]^n,
$$
for $n=3$, $5$ and $100$ using methods from recitation.
\end{enumerate}

\vfill\pagebreak



\centerline{\sc Problem Set 16: Due 27 September 2000.}

\begin{enumerate}
\item Check that $(1,1,0)$, $(0,1,1)$, and $(1,0,1)$ form a basis for
$\R^3$.  Transform this basis into an orthonormal basis using the
Gram-Schmidt algorithm.  Check that the resulting vecotrs are indeed
orthogonal!

\item Check that the vectors $(1,0,0,0)$, $(1,1,0,0)$, $(1,1,1,0)$ and 
$(1,1,1,1)$ form a basis of $\R^4$.
Use the Gram-Schmidt algorithm to make this into an
orthonormal basis.  

\item Consider the orthonormal vectors
$$
\begin{array}{ccc}
v_1=\left[\begin{array}{c}\frac{1}{\sqrt{3}}\\ \frac{1}{\sqrt{3}}\\
\frac{1}{\sqrt{3}} \end{array}\right] &
\& &
v_2=\left[\begin{array}{c}\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \\ 0
\end{array}\right]
\end{array}
$$
in $\R^3$.  Find some other vector
$$
b=\left[\begin{array}{c}b_1\\ b_2\\ b_3\end{array}\right]
$$
such that $v_1$, $v_2$, and $b$ are a basis of $\R^3$.  Then use the
Gram-Schmidt algorithm to make your basis into an orthogonal one.

\item Check that the matrix
$$
Q=\left[\begin{array}{cc}\cos(\theta) & -\sin(\theta)\\
			\sin(\theta) & \cos(\theta)
	\end{array}\right]
$$
is an orthogonal matrix by checking that $Q\cdot Q^T=I$. Also, check
that $||Q\cdot v||=||v||$ for the vector $v=\left[\begin{array}{c}
2\\1\end{array}\right]$. 

\item We have the following theorem.
\begin{thm} If $\{ v_1,\dots,v_n\}$ is an orthonormal basis for
$\R^n$, then for any $v\in\R^n$, we can write
$$
v=c_1v_1+\cdots c_nv_n,
$$
where $c_i=v\cdot v_i$ $(1\leq i\leq n)$, where $\cdot$ is the dot
product.
\end{thm}
\begin{enumerate}
\item Use this theorem to write the vector $(3,2)$ as
linear combinations of the vectors
$$
\begin{array}{ccc}
\left[\begin{array}{c} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}}
\end{array}\right] &
\& &
\left[\begin{array}{c} -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}}
\end{array}\right]
\end{array}
$$
\item Use this theorem to write $(1,2,-1)$ in terms of the basis
$$
\begin{array}{cccc}
\left[\begin{array}{c} 1\\1\\1\end{array}\right], &
\left[\begin{array}{c} 1\\1\\0\end{array}\right] &
\& &
\left[\begin{array}{c} 1\\0\\0\end{array}\right]
\end{array}
$$
\item (\sc{Optional}) Prove the above theorem.
\end{enumerate}


\item Consider the two bases of $\R^3$,
$$
B=\left\{\begin{array}{cccc}
\left[\begin{array}{c} 1\\0\\0\end{array}\right], &
\left[\begin{array}{c} 0\\1\\0\end{array}\right] &
\& &
\left[\begin{array}{c} 0\\0\\1\end{array}\right]
\end{array}\right\},
$$
and
$$
\hat{B}=\left\{\begin{array}{cccc}
\left[\begin{array}{c} 1\\1\\1\end{array}\right], &
\left[\begin{array}{c} 1\\1\\0\end{array}\right] &
\& &
\left[\begin{array}{c} 1\\0\\0\end{array}\right]
\end{array}\right\}.
$$
The change of basis matrix $M_{\hat{B}}^B$ is
$$
M_{\hat{B}}^B=\left[\begin{array}{ccc}1&1&1\\1&1&0\\1&0&0\end{array}\right].
$$
Compute $M^{\hat{B}}_B$.

\item Now consider the two bases
$$
\hat{B}=\left\{\begin{array}{cccc}
\left[\begin{array}{c} 1\\1\\1\end{array}\right], &
\left[\begin{array}{c} 1\\1\\0\end{array}\right] &
\& &
\left[\begin{array}{c} 1\\0\\0\end{array}\right]
\end{array}\right\},
$$
and
$$
\tilde{B}=\left\{\begin{array}{cccc}
\left[\begin{array}{r} 2\\-1\\0\end{array}\right], &
\left[\begin{array}{r} -1\\2\\-1\end{array}\right] &
\& &
\left[\begin{array}{r} 0\\-1\\2\end{array}\right]
\end{array}\right\}.
$$
Compute the matrices $M^{\hat{B}}_{\tilde{B}}$ and $M_{\hat{B}}^{\tilde{B}}$.
\end{enumerate}


\vfill\pagebreak



\centerline{\sc Problem Set 17: Due 28 September 2000.}

Turn in problems $1$ through $4$.  Use the rest as review for the final
exam.

\begin{enumerate}
\item Consider the matrix
$$
A=\left[\begin{array}{rr}1&0\\-1&2\end{array}\right].
$$
\begin{enumerate}
\item Find the eigenvalues of $A$.
\item Find the eigenvectors of $A$.
\item Diagonalize $A$: write it as $A=PDP^{-1}$.
\end{enumerate}


\item Consider the matrix
$$
A=\left[\begin{array}{rrr}-3&4&-4\\-3&5&-3\\-1&2&0\end{array}\right].
$$
\begin{enumerate}
\item Find the eigenvalues of $A$.  Just kidding!  The eigenvalues are
$-1$, $1$ and $2$.  Two eigenvectors are
$$
\begin{array}{ccc}
v=\left[\begin{array}{r}2\\1\\0\end{array}\right] &
\& &
w=\left[\begin{array}{r}1\\0\\-1\end{array}\right]. 
\end{array}
$$
Check that these are eigenvectors.  What are the corresponding
eigenvalues?
\item Find a third eigenvector corrseponding to the third eigenvalue.
\end{enumerate}

\item Suppose that $A$ is an $n\times n$ matrix, and that $A^2=A$.
What can you say, then, about the eigenvalues of $A$?

\item Suppose $A$ is a $3\times 3$ matrix with eigenvalues $1$, $2$
and $3$.  If $v_1$ is an eigenvector for the eigenvalue $1$, $v_2$ for 
$2$, and $v_3$ for $3$, then what is $A\cdot(v_1+v_2-v_3)$?

\end{enumerate}

\centerline{\sc Review Problems}

\noindent This is not intended to be a practice exam, but merely some
problems to review the major concepts we covered in the course.  You
should understand how to do these problems, but you should also review 
your old problem sets, exams, and lecture notes.

\begin{enumerate}

\item Graph the follwing functions.
\begin{enumerate}
\item $y=-x^3-2x^2+5x-6$
\item $y=\frac{(x-2)(x-1)}{(x+2)}$
\item $y=\sin(\theta)+\cos(\theta)$
\item $y=e^x+e^{2x}$
\end{enumerate}

\item Suppose that $f(t)$ is a function describing the position of a
particle at time $t$.  In this case, what does the derivative $f'(t)$
mean?

\item Suppose that $y=f(x)$ is a function.  What does $f'(x)$ mean
graphically?

\item Take the derivative of the following functions.
\begin{enumerate}
\item $y=\sin(\theta)\cos(\theta)$
\item $y=\sqrt{x^2+2x-3}$
\item $y=\frac{x-4}{x-2}$
\item $y=\ln(x)\cdot\sin(x)$
\end{enumerate}

\item ArsDigita University is going to build a new building.  It is to 
have floor area $3500$ square meters.  It will be a rectangle with
three solid brick walls and a glass front (with a beautiful etched
ADUni logo).  The glass costs $1.8$ times as much as the brick wall
per linear foot.  What dimensions of the building will minimize the
costs of materials for the walls and front?  (Of course, we are
ignoring all other factors in the price -- expensive roofs and floors
are no problem.  Also, we should have a place for bathrooms this
time!)

\item Compute the following integrals.
\begin{enumerate}
\item $\int\sin(x)\cos(x)\ dx$
\item $\int \frac{1}{\sqrt{3x-4}}\ dx$
\item $\int 2xe^x\ dx$
\item $\int x^3-4x-2\ dx$
\item $\int 7x^6\ln(x)\ dx$
\end{enumerate}

\item Evaluate the following definite integrals using the Fundamental
Theorem of Calculus.  

\begin{enumerate}
\item $\int_{0}^{2\pi} \cos(x)\ dx$
\item $\int_{-3}^{2} x^4+2x^3-5x^2-6x\ dx$
\item $\int_{0}^{\frac{3\pi}{2}} \sin(x)\ dx$
\end{enumerate}


\item Compute the geometric area of the following functions on the
corresponding intervals.  These are the same functions and intervals
as in the previous problem.  Note the difference between geometric and 
algebraic area!

\begin{enumerate}
\item $f(x)=\cos(x)$ on $[0,2\pi]$
\item $f(x)=x^4+2x^3-5x^2-6x=x(x-2)(x+1)(x+3)$ on $[-3,2]$
\item $f(x)=\sin(x)$ on $[0,\frac{3\pi}{2}]$
\end{enumerate}

\item Graph the following regions.  Rotate them around the $x$-axis and
compute their volumes.

\begin{enumerate}
\item The region below $f(x)=5\sqrt{x}$, above $y=0$ and to the left of 
$x=4$.
\item The region below $f(x)=9x-x^2$ above $y=0$.
\item The region below $f(x)=4x-x^2$ and above $g(x)=3(x-2)^2$.
(Hint: $f(x)=g(x)$ when $x=1,3$.)
\end{enumerate}


\item Given vectors $v=\left[\begin{array}{c}2 \\
-1\end{array}\right]$ and $w=\left[\begin{array}{c}1 \\
-4\end{array}\right]$, draw these two vectors in $\R^2$.
\begin{enumerate}
\item Write the vector $\left[\begin{array}{c}5 \\
3\end{array}\right]$ as a linear combination of $v$ and $w$.
\item Use Gram-Schmidt to make this into an orthonormal basis for
$\R^n$.
\end{enumerate}


\item Consider the following subsets of $\R^4$.  Are they subspaces?
For each subspace, write down a basis for it.  What is it's dimension?
\begin{enumerate}
\item Vectors with all four entries equal to each other.
\item Vectors with the last entry equal to 2.
\item Vectors whose entries sum up to $0$.
\item Vectors whose entries sum up to $1$.
\end{enumerate}

\item Solve the following system of equations.
$$
\left\{\begin{array}{rrrrrrr}2x&+&y&+&2z&=&3\\
			      x&-&2y&-&z&=&-2\\
			       & &4y&-&z&=&1\end{array}\right.
$$

\item Given matrices, you should be able to multiply them and find
their inverses (if they exist!).

\item Find the complete solution to
$$
\begin{array}{cccc}
\left[\begin{array}{rrrrrrr}
	1&-1&-2&-3&0&-2&-2\\
	2&-2&3&8&0&3&1\\
	-1&1&0&-1&1&2&3\\
	0&0&2&4&0&2&4
      \end{array}\right] &
\mathbf{x} & = & \left[\begin{array}{r}-1\\2\\1\\3\end{array}\right].
\end{array}
$$

\item Are the following sets of vectors bases of $\R^3$?
\begin{enumerate}
\item $(1,2,1)$ and $(-1,0,3)$.
\item $(2,4,-1)$, $(-2,0,1)$ and $(0,4,0)$.
\item $(1,2,3)$, $(-1,2,0)$ and $(2,0,0)$.
\item $(2,-1,0)$, $(2,2,4)$, $(1,-2,4)$, and $(1,-1,0)$.
\end{enumerate}

\item For the sets of vectors in the previous problem that {\em are}
bases, write down the change of basis matrices to change from the
standard basis in $\R^3$ to that particular basis, and vice versa.
Use your change of basis matrix to write $(2,-7,9)$ as a linear
combination of the basis vectors.






\end{enumerate}




\end{document}