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\begin{document}
\title{Math 241 Homework}
\author{Mary Radcliffe}
\date{Due 4 October 2018}
\maketitle

Complete the following problems. Fully justify each response.

\begin{enumerate}

\item Complete problems 4.17.8-9 on page 208-209 of Coding the Matrix.

\item Find the standard matrix for each of the following linear transformations from $\R^2\to\R^2$. Determine if each transformation is injective, surjective, both or neither. Explain your response.
\begin{enumerate}
\item Reflection over the line $y=x$
\item Reflection over the line $y=-x$
\item Projection onto the line $y=x$
\end{enumerate}


\item Let $A$ be an $m\times n$ matrix and $B$ be an $n\times p$ matrix. Prove that the $j^{\textrm{th}}$ row of $AB$ is the product of the $j^{\textrm{th}}$ row of $A$ (viewed as a $1\times n$ matrix) with $B$.


\item Determine if the following statement is true or false:
\begin{quote}Let $A, X, Y$ be $n\times n$ matrices, satisfying $AX=AY$. Then $X=Y$.\end{quote}
If the statement is true, prove it. If the statement is false, give an example showing why, and add a hypothesis that would make it true.

\item Suppose $A$ is an invertible matrix. Prove that $A^T$ is also invertible, and $(A^T)^{-1} = (A^{-1})^T$ (this is sometimes denoted by $A^{-T}$).


\item Prove that a $2\times 2$ matrix $A=\left[\begin{array}{cc}a&b\\c&d\end{array}\right]$ is invertible if and only if $ad-bc\neq 0$. (Hint: you can do this by demonstrating an inverse).


\item Complete the fourth problem set found at \url{autolab.andrew.cmu.edu}. The submission for this is directly on autolab, no need to hand it in on paper.
\end{enumerate}
\end{document}