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## Main Question or Discussion Point

Let W be a subspace of a vector space V. We define a relation v~w if v-w is an element of W.

It can be shown that ~ is an equivalence relation on V.

Suppose that V is R^2. Say W1 is a representative of the equivalence class that includes (1,0). Say W2 is a representative of the equivalence class that includes (0,1). Obviously the zero vector is related to (1,0) and (0,1).

But either two equivalence classes are similar, or they are disjoint. Am I missing something out?

It can be shown that ~ is an equivalence relation on V.

Suppose that V is R^2. Say W1 is a representative of the equivalence class that includes (1,0). Say W2 is a representative of the equivalence class that includes (0,1). Obviously the zero vector is related to (1,0) and (0,1).

But either two equivalence classes are similar, or they are disjoint. Am I missing something out?