Question

Question 10. Suppose U and W are subspaces of V, dim V = n, dim U + dim W = n, and U NW = {0}. Prove that V=UW.

Answer 1

Answer: If the dimension of V is n, then V has n elements.

Now, dim(U) + dim(W) = n, this means that the addition of the dimensions of U and W also has n elements.

and because U and W are subspaces of V, you know that every element on U and W is also an element of V.

If U ∩ W = ∅, means that there are no elements in common between U and W.

Because there are no elements in common, then Dim(U) + Dim(W) = dim(U ∪ W) = n

So U ∪ W has the same number of elements as V, and every element of W and U is also an element of V

this means that U ∪ W = V.