Solving U, V, W Subspaces Problem

[mivanova]

Hi,
I have thius problem to solve. Please, help me!

1. Prove or disprove if U, V, W are subspaces of V for which
U (dir sum) W = V (dir sum) W then U=V

Thank you in advance!

 

[Defennder]

Did you mistype the question? Does it really read V (direct sum) W ?

If it's indeed correct then think carefully about what this implies for any vector v in V and how it may be expressed as a unique sum of vectors v,w from V and W. What does it say about w?

And with this in mind, look at the left-hand side. Is this sufficient alone to conclude U=V?

 

[mivanova]

Hi,
It's indeed (direct sum) and I think that the statement is it's not true. I can't prove it though.
Thanks!

 

[Defennder]

If you can't prove it, try looking for a counter-example. Providing a single counter-example without showing why the statement isn't necessarily true would give you full marks, whereas doing the latter only gets you about half marks.

 

[Defennder]

Oh man, I can't believe I made such a stupid error. Ok, forget what I said earlier and look at V (dir sum) W. What is the subspace spanned by that, taking into account the the definition of direct sum?

What does that say about W? After you're done with this, think about the subspace spanned by U (dir sum) W, and what does it mean for U when the equality stated in the proposition holds.