Proving Subspace Equality in Vector Spaces

[kings7]

Homework Statement

I've had no problem with Chapter 1 so far, except this last proof. I arrived at an answer I thought was correct, but have been informed that it is not. This is for self-study on my part... trying to stay ahead of the curve in the upcoming fall classes. Don't laugh... I know this should be easy :(

Prove or give a counterexample: if U₁, U₂, W are subspaces of V (V being a vector space over a field) such that
V = U₁⊕W and V = U₂⊕W,
then U₁ = U₂.

The Attempt at a Solution

Given that V = U₁⊕W we know that V = U₁ ∩ W = {0} and
given that V = U₂⊕W we know that V = U₂ ∩ W = {0}. (From Theorem 1.9 in the book).

Thus U₁ ∩ W = {0} = U₂ ∩ W thus U₁ = U₂.


Any help would be appreciated. I think I'm making a logical fallacy somewhere (maybe I can't say that the two subspaces are equal using intersection of sets?) but I don't know.

Thanks!

 

[micromass]

Hi kings7!

Thus U₁ ∩ W = {0} = U₂ ∩ W thus U₁ = U₂.

This is not correct. Here's an example (of sets, not subspaces):

$$\{0,1\}\cap \{0\}=\{0\}=\{0,2\}\cap \{0\}$$

but

$$\{0,1\}\neq \{0,2\}$$

So this is why your proof is sadly incorrect!

In fact, I'd start by looking for a counterexample...

 

[kings7]

Thank you for the quick reply!

Got it. That makes it a lot more clear (I don't know what I was thinking).

I think I've found a counterexample! Thank you again for your help. :)

 

[micromass]

Subspaces are certainly considered sets! But not all sets are subspaces!

I'm interested, what is your example?

 

[kings7]

We make it so that there is something similar to the example you gave me.

Let's start by letting dim(V) = 2 and restrict it to the real numbers. In other words V = R² (since I'm not sure if something funny would happen with complex numbers).

So let's let W = {0, a} where a is an element of the reals. Let U₁ = {a, a} and U₂ = {-a, a} where -a is the additive inverse of a.

So we can see that U₁ and U₂ are still direct sums with W because U₁ ⊕ W = U₂ ⊕ W = V. However, U₁ =! U₂.

Is this correct?

 

[micromass]

Yes, I see your idea and it is correct. But I feel obliged to comment a bit about notation. Saying W=(0,a) doesn't make sense, you need to say that W is the space spanned by (0,a), or say that W=<(0,a)> or whatever notation you use.
Also, don't forget to say that a can't be zero!

Well done!

 

[kings7]

Ahh! You're right. Of course, I meant the spanning set but I was being notationally lazy (never an excuse, I know). :)

But the zero thing I have to remember for the future. Thank you again! Getting through this book is my summer project, and I expect to have many questions in the future. Good to know there are people like you around!