4.1 Prove Proposition 4.1.6.

4.2 For each of the following sets, determine whether it is a vector space over the given finite field F _q. If it is a vector space, determine the number of distinct bases it can have.

1. q = 2, S = {( a, b, c, d, e) : a + b + c + d + e = 1},

2. q = 3, T = {( x, y, z, w) : xyzw = 0},

3. q = 5, U = {( ? + ?, 2 ?, 3 ? + v, v) : ?, ?, v ? F ₅},

4. q prime, V = {( x ₁, x ₂, x ₃) : x ₁ = x ₂ ? x ₃}.

4.3 For any given positive integer n and any 0 ? k ? n, determine the number of distinct subspaces of of dimension k.

4.4

1. Let F _q be a subfield of F _r. Show that F _r is a vector space over F _q, where the vector addition and the scalar multiplication are the same as the addition and multiplication of the elements in the field F _r, respectively.

2. Let ? be a root of an irreducible polynomial of degree