Equivalence class and linear subspace

(Białynicki-Birula, Algebra liniowa z geometrią, str 55)

Assume that we have a linear space Z³₃ and let V be a set of three vectors: {(0,0,0),(1,0,1), (2,0,2)} .

Prove that:
1) V is a linear subspace of Z³₃
2) There are exactly 9 equivalence classes of V.

Explanation:

Z₃ is a set {0, 1, 2} having operations defined in the set of integers modulo 3. Z₃ is a field.

Z³₃ is a set of all 3-tuples (ordered) of Z₃. Examples: (1,1,2), (0,1,2) etc. Z³₃ is a linear space over Z₃.

Equivalence class: http://en.wikipedia.org/wiki/Quotient_space_(linear_algebra)

Proof:

It is very easy to prove 1).

Let's look at 2).

Equivalence class (Polish: warstwa) is a set of all vectors being the sum of any vector of Z³₃ minus V and all vectors belonging to V.

Example:
Consider the vector (1,0,0). The equivalence class defined by this vector (tuple) is: {(1,0,0), (2,0,1), (0,0,2)}.

Every equivalence class is defined by a vector (tuple). So what we have to do is to find the minimal set E (subset of Z³₃) that every vector of Z³₃ is the sum of one vector of E and one vector of V.

Let Z1 be a set of all vectors (a,b,c) where a != c. Forget for a moment about b.

Let V1 be a set {(0,0), (1,1), (2,2)}. It is clear that every vector (a,c) is the sum of one of the vector of V1 and (0,1) or (0,2).

For instance:
(2,0) = (2,2) + (0,1) (modulo 3 arithmetic) (0,2) = (0,0) + (0,2) etc.

So, with respect to b, we have a set E1 of six vectors = { (0,0,1),(0,0,2),(0,1,1),(0,1,2),(0,2,1),(0,2,2)} and it defines all vectors of Z1.

Let Z2 be a set of vectors (a,b,c) where a=c and vector does not belong to V . It is clear that a set E2 of three vector = { (0,1,0), (0,2,0), (2,0,2) } is enough to define every vector belonging to Z2.

For instance:
(2,1,2) = (2,0,2) + (0,1,0)
(1,0,1) = (1,0,1) + (2,0,2)

So set E = E1 + E2 (containing 9 vectors) defines Z1 + Z2.

QAD.