Chapter 3.1, Problem 53E

In Exercises 53 and 54, we will work with the binary digits (or bits) 0 and 1, instead of the real numbers $ℝ$ . Addition and multiplication in this system are defined as usual. except for the rule $1+1=0$ . We denote this number system with ${\mathbb{F}}_2$ , or simply $\mathbb{F}$ . The set of all vectors with n components in $\mathbb{F}$ is denoted by ${\mathbb{F}}^n$ ; note that ${\mathbb{F}}^n$ consists of $2^n$ vectors. (Why?) In information technology, a vector in ${\mathbb{F}}^8$ is called a byte. (A byte is a string of eight binary digits.)
The basic ideas of linear algebra introduced so far (for the real numbers) apply to $\mathbb{F}$ without modifications.

A Humming matrix with n rows is a matrix that contains all nonzero vectors in ${\mathbb{F}}^n$ as its columns (in any order). Note that there are $2^n-1$ columns. Here is an example:

$H=\left[\begin{array}{ccccccc}1& 0& 0& 1& 0& 1& 1\\ 0& 1& 0& 1& 1& 0& 1\\ 0& 0& 1& 1& 1& 1& 0\end{array}\right],\begin{array}{c}3\text{rows}\\ 2^3-1=7\text{columns}\text{.}\end{array}$
a. Express the kernel of H as the span of four vectors in F of the form
${\stackrel{\to }{v}}_1=\left[\begin{array}{l}*\\ *\\ *\\ 1\\ 0\\ 0\\ 0\end{array}\right],{\stackrel{\to }{v}}_2=\left[\begin{array}{l}*\\ *\\ *\\ 0\\ 1\\ 0\\ 0\end{array}\right],{\stackrel{\to }{v}}_3=\left[\begin{array}{l}*\\ *\\ *\\ 0\\ 0\\ 1\\ 0\end{array}\right],{\stackrel{\to }{v}}_4=\left[\begin{array}{l}*\\ *\\ *\\ 0\\ 0\\ 0\\ 1\end{array}\right].$

b. Form the $7\times 4$ matrix $M=\left[\begin{array}{cccc}|& |& |& |\\ {\stackrel{\to }{v}}_1& {\stackrel{\to }{v}}_2& {\stackrel{\to }{v}}_3& {\stackrel{\to }{v}}_4\\ |& |& |& |\end{array}\right]$ .

Explain why $\text{im}\left(M\right)=\ker \left(H\right)$ . If $\stackrel{\to }{x}$ is an arbitrary vector in ${\mathbb{F}}^4$ , what is $H\left(M\stackrel{\to }{x}\right)$ ?