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\begin{document}
\title{Math 241 Homework}
\author{Mary Radcliffe}
\date{Due 13 September 2018}
\maketitle

Complete the following problems. Fully justify each response.

\begin{enumerate}

\item Read Sections 4.7.3-4.7.5 in Coding the Matrix, about error-correcting codes. Using the Hamming Code and matrix $H$ as described in Section 4.7.5, answer the following questions:
\begin{enumerate}
\item Assuming that $\vec{e}$ has at most one bit error, and that $H\vec{e}=\left[\begin{array}{c}1\\1\\0\end{array}\right]$, what is $\vec{e}$?
\item Suppose that the vector \[\tilde{\vec{c}} = \left[\begin{array}{c}1\\0\\1\\0\\1\\1\\1\end{array}\right]\] is received over a noisy channel that introduces at most one bit error, and $H$ is used to decode it. What was the original message intended to be?
\end{enumerate}

\item Consider the matrix 
\begin{equation*}
A = \left[\begin{array}{cc}1 & -2 \\ 0 & 0\end{array}\right].
\end{equation*}
Draw a picture of the null space of $A$ in $\R^2$. On the same graph, draw a picture of the solution set to the equation $A\vec{x} = \vec{b}$, where $\vec{b} = \left[\begin{array}{c} 3 \\ 0\end{array}\right]$.


\item Consider the matrix 
\begin{equation*}
A = \left[\begin{array}{cccccc} 1 & 2 & -3 & -1 & 2 & 1\\ 0 & 0 & 1 & -1 & 1 & 0\\ 0 & 0 & 0 & 1 & 0 & 1\\ 0 & 0 & 0 & 0 & 1 & 1\end{array}\right].
\end{equation*}
\begin{enumerate}
\item Write the null space of $A$ as the span of some list of vectors.
\item Consider the equation $A\vec{x} = \vec{b}$, where $\vec{b}=\left[\begin{array}{c}0\\0\\1\\1\end{array}\right]$. Find one solution to this equation (you may do this by observation, and then demonstrate your solution is correct). Use the null space to write an expression for all solutions.
\end{enumerate}

\item Complete Exercises 4.10.7 and 4.10.8 on pages 176-177 in Coding the Matrix.


\item Recall, from lecture, that the set of polynomials having degree at most $n$ with coefficients from $\R$ is a vector space. Denote this vector space by $\mathcal{P}_n(\R)$. Define a function $D:\mathcal{P}_n(\R)\to\mathcal{P}_{n-1}(\R)$ by $D(p(x)) = \frac{d}{dx}p(x)$. 
\begin{enumerate}
\item Show that this function is linear.
\item Describe the kernel and range of this function.
\end{enumerate}

\item Let $f:\R^n\to \R^m$ be a function, and let $k$ be any (fixed) positive integer with $k\geq 2$. Prove that $f$ is linear if and only if \[f(a_1v_1+a_2v_2+\dots+a_kv_k)=a_1f(v_1)+a_2f(v_2)+\dots+a_kf(v_k)\] for all $a_1, a_2, \dots, a_k\in \R$ and $v_1, v_2, \dots, v_k\in \R^n$.

\item Complete the third 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}