Vector Space Problem

[cubixguy77]

Prove or give a counterexample to the following: if U, W, X are all subspaces of a vector space V, such that
U + W = X + W
then U = X

Prove or give a counterexample: if U, W, X are subspaces of a vector space V, such that
V = U ⊕ W and
V = X ⊕ W
then U = X

i cannot think of any counterexamples to either, but i'm not sure how to proceed with the proofs themselves. can anyone help me out? thanks.

[HallsofIvy]

Prove or give a counterexample to the following: if U, W, X are all subspaces of a vector space V, such that
U + W = X + W
then U = X
This is pretty obviously not true. Take V to be R^3, U= <x, y, 0>, V= <0, 0, z>, X= R^3.

Prove or give a counterexample: if U, W, X are subspaces of a vector space V, such that
V = U ⊕ W and
V = X ⊕ W
then U = X
Unfortunately, you have a "special symbol" in there that my reader won't show properly. It can't be intersection because then it couldn't be equal to V. It can't be union because unions are not, in general subspaces.

i cannot think of any counterexamples to either, but i'm not sure how to proceed with the proofs themselves. can anyone help me out? thanks.

[cubixguy77]

the symbols are direct sums (circle with plus sign in it)

[HallsofIvy]

For some reason, now I can see it! The important difference is that the direct sum of subspaces requires that the subspaces have only the 0 vector in common, so my counterexample in the first problem doesn't work here.

Specifically, if you write a basis for U and a basis for W, then their union is a basis for U⊕W. Of course the same is true for U and X. You should be able to use that to prove this is true.