\documentclass[12pt]{amsart} \setlength{\parsep}{3pc} \setlength{\itemsep}{0.2in} \usepackage{fullpage} \usepackage{epsfig} \newcommand{\Z}{\mathbb Z} \newcommand{\F}{\mathbb F} \newcommand{\R}{\mathbb R} \newcommand{\C}{\mathbb C} %\newcommand{\to}{\rightarrow} \title[Problem Sets 10--14]{Problem Sets 10 to 14. Due as specified.} \begin{document} \maketitle %\renewcommand{\arraystretch}{1.5} \centerline{\sc Problem Set 10: Due 19 September 2000.} \noindent {\bf Reading.} {\em Matrices and Transformations}, pp. 1--12. \noindent {\bf Supplementary reading.} Strang, Chapter 1 and Section 2.1. \begin{enumerate} \item Draw $v=\left[\begin{array}{c}2 \\ 1\end{array}\right]$ and $w=\left[\begin{array}{c}1 \\ 3\end{array}\right]$, along with $v+w$, $2v+w$, and $v-w$ in one $xy$-plane. \item The vectors $v=\left[\begin{array}{c}1 \\ 2\end{array}\right]$ and $w=\left[\begin{array}{c}4 \\ -2\end{array}\right]$ are perpendicular. Thus, $v$, $w$ and $v+w$ form a right triangle. Check the Pythagorean Theorem, $||v||^2+||w||^2=||(v+w)||^2$ in terms of the definition of length $||v||$. \item To save space, I will write column vectors as rows. For $u=(0,1,2)$, $v=(1,3,0)$, and $w=(1,0,4)$, find $||u||$, $||v||$, $||w||$, $u\cdot v$, $u\cdot w$, and $v\cdot w$. Check the Law of Cosines for $u$ and $v$, as well as the Schwartz inequality for $v$ and $w$. \item Solve the systems of equations $$ \begin{array}{ccc} \left\{\begin{array}{c}2x+y=5 \\ x-3y=6\end{array}\right. & \phantom{boo!} & \left\{\begin{array}{c}x+y-z=2 \\ x-y+2z=1 \\ y+4z=0\end{array}\right. \end{array} $$ \item Let $A$, $B$, $C$, $D$, $E$ and $F$ be the matrices below. Find $B+D$, $2E-F$, $AC$, $BC$, $CB$, $ACD$, $EF$, $FE$ and $CEF$. In particular, note that $EF\neq FE$! $$ \begin{array}{ccc} A=\left[\begin{array}{ccc}1&2&3\\0&1&2\\1&-1&0\end{array}\right] & B=\left[\begin{array}{ccc}1&2&-2\\0&-1&4\end{array}\right] & C=\left[\begin{array}{cc}1&0\\2&1\\3&-1\end{array}\right] \\ & & \\ D=\left[\begin{array}{ccc}0&2&1\\2&1&-1\end{array}\right] & E=\left[\begin{array}{cc}2&4\\3&1\end{array}\right] & F=\left[\begin{array}{cc}1&0\\2&-1\end{array}\right] \end{array} $$ \item Draw the row and column pictures for $$ \begin{array}{l}2x-y=3\\x+y=1\end{array} $$ \item If you have $5$ linear equations in $3$ unknowns, then the row picture shows five \underline{\phantom{tlanes tlanes}}. The column picture is in what dimensional space? The equations will have a solution only if the vector on the right hand side is a combination of what? \item Consider the matrix $$ \begin{array}{ccc} A & = & \left[\begin{array}{cc}0.8&0.3\\0.2&0.7\end{array}\right] \end{array}. $$ Compute $A^2$, $A^3$ and $A^4$. What do you notice about the columns? \item What matrix sends $v=(1,0)$ to $(0,1)$ and also sends $w=(0,1)$ to $(-1,0)$? This matrix rotates $\R^2$ by $90^{\circ}$. \end{enumerate} \vfill\pagebreak \centerline{\sc Problem Set 11: Due 20 September 2000.} \noindent {\bf Reading.} {\em Matrices and Transformations}, sections 2-2 and 2-5. \noindent {\bf Supplementary reading.} Strang, sections 2.1--2.4. \begin{enumerate} \item Solve the followng systems of equations using Gaussian elimination. $$ \begin{array}{ccc} \left\{\begin{array}{rrrrrrr}3x&+&y&+&z&=&6\\ x&-&y&-&z&=&-2\\ & &4y&+&z&=&3\end{array}\right. & \phantom{boo} & %Makes a space of length `boo' \left\{\begin{array}{rrrrrrr}x&+&2y&-&z&=&3\\ 3x&-&y&+&2z&=&3\\ 2x&+&y&+&4z&=&1\end{array}\right. \end{array} $$ \item Consider the matrix $$ \left[\begin{array}{rrr}1&2&4\\-1&-3&-2\\0&1&c\end{array}\right]. $$ For what value(s) of $c$ can you not perform elimination (without row exchanges) on this matrix? \item Consider the system of equations, $$ \left\{\begin{array}{rrrrrrr} x&+&y&+&2z&=&1\\ 2x&+&2y&-&z&=&1\\ & &y&+&cz&=&2 \end{array}\right. $$ For what values of $c$ does this system have no solutions? one solution? infinitely many solutions? \item Compute the inverses of the following matrices, using elimination. $$ \begin{array}{ccc} \left[\begin{array}{rrr}3&1&2\\-1&0&1\\0&1&6\end{array}\right] & \phantom{boooo!} & \left[\begin{array}{rrr}1&2&1\\-1&4&-2\\1&3&1\end{array}\right] \end{array} $$ \item Compute the inverse of $$ A=\left[\begin{array}{rrr}a&b&b\\a&a&b\\a&a&a\end{array}\right]. $$ For what $a$ and $b$ is there no inverse? \item Compute the inverse of the general $2\times 2$ matrix $$ \left[\begin{array}{rr}a&b\\c&d\end{array}\right]. $$ What condition on $a$, $b$, $c$, and $d$ ensures that this will exist? \item Suppose $Ax=b$ has two solutions $v$ and $w$ (with $v\neq 2$). Then show that $\frac{1}{2}(v+w)$ is also a solution, although $v+w$ is not. \item In recitation, we saw that for a permutation matrix, $P$, $PA$ has the same entries as $A$, but the rows are permuted around. What would you guess happens if we multiply in the other order, that is what does $P$ do when we multiply $AP$? Check your answer in the $2\times 2$ case. \end{enumerate} \vfill\pagebreak \centerline{\sc Problem Set 12: Due 21 September 2000.} \noindent {\bf Reading.} {\em Matrices and Transformations}, sections 1-3, 1-4, and 2-1. \noindent {\bf Supplementary reading.} Strang, sections 2.5--2.6, Chapter 5. \begin{enumerate} \item Factor the following matrices into the form $A=L\cdot U$. $$ \begin{array}{ccc} \left[\begin{array}{rrr}1&0&1\\1&2&3\\0&2&4\end{array}\right] & \phantom{booooo!} & \left[\begin{array}{rrr}1&3&2\\2&8&4\\3&13&7\end{array}\right] \end{array} $$ \item Factor the following symmetric matrices into $A=L\cdot D\cdot L^T$. $$ \begin{array}{ccc} \left[\begin{array}{rrrr}1&1&1&1\\1&2&3&4\\1&3&6&10\\1&4&10&20\end{array}\right] & \phantom{boo!} \left[\begin{array}{rrrr}1&1&1&1\\1&2&2&2\\1&2&3&3\\1&2&3&4\end{array}\right] \end{array} $$ \item Given a permutation matrix $$ \begin{array}{ccc} P & = & \left[\begin{array}{rrrr}0&0&1&0\\0&1&0&0\\0&0&0&1\\1&0&0&0\end{array}\right] \end{array} $$ what is $P^{-1}$? Do you recognize this matrix? \item There are only finitely many ($n!$) $n\times n$ permutation matrices. Use this fact to show that $P^r=I$ for some $r$. \item For what value(s) of $c$ can you {\em not} factorize $A=L\cdot U$? $$ \begin{array}{ccc} A& =&\left[\begin{array}{rrr}1&2&4\\2&c&7\\0&1&3\end{array}\right] \end{array} $$ \item Compute the determinants of the following matrices. $$ \begin{array}{ccccc} \left[\begin{array}{rr}1&2\\-4&3\end{array}\right] & \phantom{boo} & \left[\begin{array}{rrr}1&-1&0\\0&1&2\\2&1&-2\end{array}\right] & \phantom{boo} & \left[\begin{array}{rrrr}1&0&-1&0\\0&1&0&0\\0&0&1&2\\2&0&1&-2\end{array}\right] \end{array} $$ \item Prove that $\det(A^{-1})=\frac{1}{\det(A)}$. \item Let $$ \begin{array}{ccc} A& =& \left[\begin{array}{rrrrr}1&1&1&\cdots&1\\0&2&2&\cdots&2\\0&0&3&\cdots&3\\ \vdots&&&\ddots &\\ 0&0&0&\cdots&n\end{array}\right] \end{array}. $$ Compute $\det(A)$. \item If you have a block matrix $$ \begin{array}{ccc} A& =& \left[\begin{array}{rr}B&\mathbf{0}\\ \mathbf{0}&C\end{array}\right] \end{array}, $$ what is $\det(A)$? \end{enumerate} \vfill\pagebreak \centerline{\sc Problem Set 13: Due 22 September 2000.} \noindent {\bf Reading.} {\em Matrices and Transformations}, none. \noindent {\bf Supplementary reading.} Strang, sections 3.1--3.2. \begin{enumerate} \item Consider the set $M_2=\{ 2\times 2\mbox{ matrices}\}$ as a vector space. Let $$ \begin{array}{cc} A=\left[\begin{array}{rr}2&0\\0&0\end{array}\right] & B=\left[\begin{array}{rr}0&0\\0&-3\end{array}\right] \end{array} $$ \begin{enumerate} \item Name a subspace containing $A$ but not $B$. \item Name a subspace containing $B$ but not $A$. \item Is there a subspace containing $A$ and $B$ but not the $2\times 2$ identity matrix? \end{enumerate} \item Consider $\R^2$ as a vector space. Which of the following are subspaces and which are not? If not, why not? \begin{enumerate} \item $\{ (a,a^2\ |\ a\in\R\}$ \item $\{ (b,0)\ |\ b\in\R\}$ \item $\{ (0,c)\ |\ c\in\R\}$ \item $\{ (m,n)\ |\ m,n\in\Z\}$ \item $\{ (d,e)\ |\ d,e\in\R,\ d\cdot e=0\}$ \item $\{ (f,f)\ |\ f\in\R\}$ \end{enumerate} \item Show that for some $b\neq 0$, the solution set $\{ x\ |\ Ax=b\}$ does {\bf not} form a subspace. (Hint: look at Problem set $11$, problem number $7$.) \item Considerthe set $M_n=\{ n\times n\mbox{ matrices}\}$ as a vector space. Which of the following are subspaces? \begin{enumerate} \item The symmetric matrices, $S=\{ A\ |\ A^{\mathrm{T}}=A\}$ \item The non-symmetric matrices, $NS=\{ A\ |\ A^{\mathrm{T}}\neq A\}$ \item The {\em skew-symmetric} matrices, $S=\{ A\ |\ A^{\mathrm{T}}=-A\}$ \end{enumerate} \item Describe the column spaces of the following matrices. $$ \begin{array}{cc} C=\left[\begin{array}{rr}1&2\\2&0\\-1&3\end{array}\right] & D=\left[\begin{array}{rrr}1&3&2\\2&2&0\\-1&2&3\end{array}\right] \end{array} $$ \item Describe the null-space for the following matrices. $$ \begin{array}{ccc} E=\left[\begin{array}{rrr}1&2&-2\\2&1&0\end{array}\right] & F=\left[\begin{array}{rr}1&-1\\0&2\end{array}\right] & G=\left[\begin{array}{rrr}1&2&-4\\-1&1&3\\1&5&-5\end{array}\right] \end{array} $$ \item Let $P$ be the plane in $\R^3$ defined by the equation $$ x-y-z=3. $$ Find two vectors in $P$ and show that their sum is not in $P$. \item \begin{enumerate} \item Find a subset $W\subseteq \R^2$ where, for $v,w\in W$, $v+w\in W$, but $cv$ is not necessarily in $W$. \item Find a subset $W\subseteq \R^2$ where, for $v,w\in W$, $cv\in W$, but $v+w$ is not necessarily in $W$. \end{enumerate} \item Let $A$ and $B$ be any $n\times n$ matrices. If $v\in N(B)$, show that $v\in N(A\cdot B)$. If $A$ is invertible, show that if $v\in N(A\cdot B)$, then $v\in N(B)$. \end{enumerate} \vfill\pagebreak \centerline{\sc Problem Set 14: Due 25 September 2000.} \noindent {\bf Reading.} {\em Matrices and Transformations}, sections 2-4 and 2-5. \noindent {\bf Supplementary reading.} Strang, sections 3.3--3.5. \begin{enumerate} \item Find the reduced row echelon form of the following matrices. $$ \begin{array}{cc} A=\left[\begin{array}{rrrr}1&1&1&1\\1&1&1&1\\0&1&2&3\\0&1&2&3\end{array}\right] & B=\left[\begin{array}{rrr}1&2&1\\2&2&2\\1&0&1\end{array}\right] \\ & \\ C=\left[\begin{array}{rrrrr}1&2&1&2&1\\2&1&2&1&2\\0&1&0&1&0\end{array}\right] & D=\left[\begin{array}{rrr}1&2&3\\2&3&4\end{array}\right] \end{array} $$ \item Compute the ranks of the following matrices. $$ \begin{array}{cc} E=\left[\begin{array}{rrrr}1&2&0&5\\2&3&1&4\\-1&-1&-1&1\end{array}\right] & F=\left[\begin{array}{rrr}1&2&1\\-1&3&4\\2&-1&-3\end{array}\right] \\ & \\ G=\left[\begin{array}{rrr}1&2&1\\0&3&1\\-2&1&4\end{array}\right] & H=\left[\begin{array}{rr}1&3\\2&-1\\-1&-3\end{array}\right] \end{array} $$ \item Find the complete solution to the equation $$ \begin{array}{cccc} \left[\begin{array}{rrrrr} 1&3&2&4&-3\\ 2&6&0&-1&-2\\ 0&0&6&2&-1\\ 1&3&-1&4&2 \end{array}\right] & \left[\begin{array}{c}x_1\\x_2\\x_3\\x_4\\x_5\end{array}\right] & = & \left[\begin{array}{c}-7 \\ 0 \\ 12 \\ -6\end{array}\right] \end{array} $$ \item Find a matrix with the following property, or say why you cannot have one. \begin{enumerate} \item The complete solution to $B\cdot x=\left[ \begin{array}{r}1\\2\\4\end{array}\right]$ is $x=\left[ \begin{array}{r}1\\0\end{array}\right]$. \item The complete solution to $C\cdot x=\left[ \begin{array}{r}5\\1\end{array}\right]$ is $x=\left[ \begin{array}{r}1\\4\\-2\end{array}\right]$. \end{enumerate} \item The complete solution to \begin{eqnarray*} A\cdot x& =& b \end{eqnarray*} is \begin{eqnarray*} x&=&\left[\begin{array}{c}1\\0\\0\end{array}\right] + c_1\left[\begin{array}{c}0\\1\\0\end{array}\right] + c_2\left[\begin{array}{c}0\\0\\1\end{array}\right] \end{eqnarray*} What is $A$? \item Consider the set $P_2=\{ ax^2+bx+c\ |\ a,b,c\in\R\}$ of all polynomials of degree less than or equal to $2$. \begin{enumerate} \item Show that this is a vector space. \item Show that the function $$ L(p(x))=\int_0^1p(x)\ dx $$ is a linear transformation $L:P_2\to\R$. \item What is the null sapce of $L$? (That is, what polynomials does $L$ map to 0?) \item What is the range of $L$? \end{enumerate} \item \begin{enumerate} \item Mike, Shai, and Tara all decide that they are unhappy with the color scheme at ADUni, and they do something about it. They go down to the paint store, and each buy some paint. Mike wants to paint the lab aqua, so he buys one gallon of red paint, six gallons of blue paint, and one gallon of yellow paint. He spends \$44.00. Shai, on the other hand, wants to paint the front office green. He buys no red paint, two gallons of blue paint and three gallons of yellow paint. He spends \$24.00. Tara finally decides to paint the classroom purple. She buys one gallon of red paint and five gallons of blue paint, and spends \$33.00. How much does each color of paint cost? \item What is wrong with your answer to the previous problem? When Mike, Shai and Tara compare receipts, they realize that one of them was charged \$4.00 too little. Who was it? \end{enumerate} \item Heather and Tony decide to start a cookie business. Heather is going to contribute chocolate chip cookies and Tony is going to make Tony's Special Secret Recipe cookies. Heather's cookies take $\frac{3}{4}$ of an hour of preparation time, and one hour in the oven (to make 100 cookies). Tony's Special Secret Recipe cookies take one hour of prep time and a full two hours in the oven (for 100 cookies). Combined, Heather and Tony are willing to put in $30$ hours of prep time, and $50$ hours of oven time. They figure they can make \$60.00 on each $100$ of Heather's cookies and \$90.00 on each 100 of Tony's cookies. How many cookies of each type do they make in order to maximize profits? ({\sc Hint:} Set up a linear programming problem like Shai showed in recitation, and solve it geometrically by graphing the constraints and determining on which extreme values the profits are maximal.) \end{enumerate} \end{document}