PDA

View Full Version : Vector Space Problem

cubixguy77
02-05-2008, 11:15 PM
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
02-06-2008, 09:24 PM
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
02-07-2008, 05:23 AM