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\begin{document}
\title{Math 241 Homework}
\author{Mary Radcliffe}
\date{Due 6 September 2018}
\maketitle

Complete the following problems. Fully justify each response.

\begin{enumerate}

\item Complete problems 2.14.4, 2.14.7 on page 111 of Coding the Matrix. (Note: Klein uses bitstrings to represent vectors over $\Z_2$. So for example, $0011$ would denote the vector $\left[\begin{array}{c}0\\0\\1\\1\end{array}\right]$.)

\item Complete problems 3.8.4, 3.8.5 on page 146 of Coding the Matrix.

\item Use ideas discussed in class to solve the following upper triangular system of linear equations:
\begin{equation*}
\begin{array}{ccccccccc}
x_1 & + & 3x_2 & - & 2x_3 & + & 2x_4 & = & -5\\
&& 3x_2&+&4x_3&-&x_4& =& 1\\
&&&& 2x_3&+&x_4& =& -1\\
&&&&&& -5x_4 & = & 10
\end{array}
\end{equation*}

\item (ALA Problem 1.10) The record for each student in a class is given as a 10-vector $\vec{r}$, where $r_1, r_2, \dots, r_8$ are the scores for the 8 homework assignments, each on a 0-10 scale, $r_9$ is the midterm exam grade on a 0-120 scale, and $r_{10}$ is the final exam score on a 0-160 scale. The student's total course score $s$, on a 0-100 scale, is based on a weighted average of 25\% homework, 35\% midterm, and 40\% final exam. Determine a vector $\vec{w}$ so that $s=\vec{w}\cdot\vec{r}$. 

\item In lecture, we proved that vector addition is associative. Prove that vector addition is also commutative; that is, prove that $\vec{u}+\vec{v}=\vec{v}+\vec{u}$. For the purposes of this problem, you may assume that your vectors are over $\R$, and you may use standard arithmetic rules in $\R$.

\item (Similar to ALA Problem 1.18) Suppose that each of the vectors $\vec{b}_1, \vec{b}_2, \dots, \vec{b}_k$ is a linear combination of the vectors $\vec{a}_1, \vec{a}_2, \dots, \vec{a}_m$, and that $\vec{c}$ is a linear combination of $\vec{b}_1, \vec{b}_2, \dots, \vec{b}_k$. Then $\vec{c}$ is a linear combination of $\vec{a}_1, \vec{a}_2, \dots, \vec{a}_m$. Show this for the case that $m=2$. Explain why it would also be true if $m$ is larger. (The proof is not hard, but the notation is a mess)

\item Read through Coding the Matrix 2.9.4-2.9.7. Here, you will find a discussion of dot products over $\Z_2$, and how we can use $\Z_2$ for authentication schema.

\item Complete the first 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}