Question

What does condition b mean? Is it saying that we get V when we combine all elements of all subspaces?. Thanks.

Answer 1

When you add vector spaces, you get a new vector space containing all possible sums of vectors within each space. As a concrete example, if U was the subspace of R^(3) of all vectors of the form (x, 0, 0) and W is the subspace of all vectors of the form (0, y, 0), then U + W would be the set of all vectors of the form (x, 0, 0) + (0, y, 0) = (x, y, 0). That means that every vector in U + W can be written as a vector in U plus a vector in W.

Answer 2

V1 + V2 + ... + Vk = {u1 + u2 + ... + uk : each ui is in  Vi}