\documentclass{amsart}


\title{Matrix theory autumn 2011: homework 1}

\author{James Cummings}


\begin{document}

\maketitle
    
   Homework due at start of class on Wed 7 September. 



\section{Sets and logic}


\begin{enumerate}


\item  Using truth tables or otherwise, prove that:

\begin{enumerate}

\item  ``A implies B'' is logically equivalent to ``not-A or B''.

\item  ``not-(A or B)'' is logically equivalent to ``not-A and not-B''.

\item  ``not-(A implies B)'' is logically equivalent to ``A and not-B''.
       
\end{enumerate}


\item  Recall that the basic operations on sets are intersection, union and set difference.
       Let $X$ be a set, let $\phi(x)$ and $\psi(x)$ be properties that elements $x \in X$
       may have. Let $A = \{ x \in X: \phi(x) \}$ and let $B = \{ x \in X : \psi(x) \}$.
       Describe (with proof) each of the following sets in terms of the basic operations:

\begin{enumerate}

\item  $\{ x \in X : \text{not  $\phi(x)$} \}$.

\item  $\{ x \in X : \text{$\phi(x)$ or $\psi(x)$} \}$.

\item  $\{ x \in X : \text{$\phi(x)$ implies $\psi(x)$} \}$.
 
\end{enumerate} 


   For example: $\{ x \in X : \text{$\phi(x)$ and $\psi(x)$} \} = A \cap B$. 

\end{enumerate}








\section{Linear equations}


\begin{enumerate}

\item   Consider  linear systems in two variables $x,y$  of the general form
\begin{eqnarray*}
    x + 2 y & = & 1 \\
    a x + y & = & b \\
\end{eqnarray*}
    where $a, b$ are constants. 

    For which values of $a, b$ does this system have

\begin{enumerate}

\item  Exactly one solution?

\item  No solutions?

\item  Infinitely many solutions? 

\end{enumerate} 


\end{enumerate} 





\section{Matrices and vectors}

\begin{enumerate}

\item  A function $T$ from ${\mathbb R}^2$ to ${\mathbb R}^2$ is called a  {\em linear transformation} if there exists a 
       $2 \times 2$ matrix $A$ such that for all $x$ and $y$ we have $T(x, y) = (x_1, y_1)$, where  
\[
\left ( 
  \begin{array}{c}
x_1 \\ 
y_1 \\
\end{array}
\right )
= A
\left ( 
  \begin{array}{c}
x \\ 
y \\
\end{array}
\right )
\]

\begin{enumerate}

\item  Prove that if $T$ is a linear transformation then there is exactly one matrix $A$ as above. 

\item  Prove that a rotation by $\phi$ radians about the origin $(0, 0)$ is a linear transformation
     and describe the corresponding matrix. 

\item  Prove that the translation $(x, y) \mapsto (x+1, y+1)$ is {\em not} a linear transformation. 

\item  Consider the line $L$ with equation $y = a x$. Is reflection in $L$ a linear transformation?
      If yes find the corresponding matrix. 
 
\item Let $A_\phi$ be the matrix for rotation by $\phi$. Prove that $A_\phi A_\psi = A_{\phi + \psi}$. 

\end{enumerate} 


\item   Let $X$ be the set of $2 \times 2$ real matrices.  Prove or give a counterexample to
      each of the following statements:

\begin{enumerate}

\item  $A + B = B + A$ for all $A, B \in X$.

\item  $A B = B A$ for all $A, B \in X$.

\item  $A^2 = 0$ implies $A =0$ for all $A \in X$.

\item  $A A^2 = A^2 A$ for all $A \in X$. 


\end{enumerate} 


\section{Maple} 

    This section is {\em not for credit}, {\em you may collaborate} and {\em you need not hand anything in.}
    But you still should do it because probably none of this  will be true of the Maple sections in later
    homeworks. 
 


\begin{enumerate}

\item Either install Maple (CMU has a license)  on your personal computer or find a public workstation
   which has it installed. 


\item  Start Maple. 

\item  Open up a new Maple worksheet.

\item  Load the ``LinearAlgebra'' package with the command \texttt{with(LinearAlgebra);}

\item  Locate the online help for the LinearAlgebra package.

\item  Create matrices 
\[
A = \left (
\begin{array}{cc}
1 & 2 \\
4 & 5 \\
\end{array}
\right ),
B = \left (
\begin{array}{cc}
3 & 7 \\
6 & 8 \\
\end{array}
\right ).
\]

\item   Use Maple to compute $A + B, A B, B A,  A^{100}$.

\item Export your results to a PDF file. 

\end{enumerate}



\end{enumerate} 




\end{document}



