HW Number 2
Due on March 11th
 Find a quadratic polynomial \(f \in \mathbb{F}_7[x] \) such that
$$ f \equiv x^5 + 3x^4 +5x + 1 \mod{(x^3 + x + 1)} $$

Let \(G\) be the group formed by all rigid motion symmetries of an equilateral triangle (imagine a solid metal triangle with corners labelled A,B,C for instance).
 Describe the elements of \(G\).
 Is this group commutative, that is \(a * b = b * a\) for all \(a,b \in G\)?
 Is this group somehow equivalent to the group of all permutations of three items?
 Does this group have a generator?
 Let \(H = \{ (0, 0, 0, 0, 0, 0), (1, 1, 1, 0, 0, 0), (0, 0, 0, 1, 1, 1), (1, 1, 1, 1, 1, 1)\} \)
be a subgroup of \(\mathbb{F}_2^6\) under vector addition modulo 2. Find all of the cosets of \(H\).
Find the element of each coset with smallest Hamming weight.
 Find a basis of the nullspace of
$$ \left(\begin{array}{rrrrr}
0 & 1 & 0 & 0 & 1 \\
1 & 0 & 0 & 1 & 1 \\
0 & 0 & 1 & 0 & 1
\end{array}\right) $$
over \(\mathbb{F}_2\).