Proof involving subsets of a vector space

[zapz]

Homework Statement

This is a problem from chapter 1.3 of Linear Algebra by F/I/S.

Let W_{1} and W_{2} be subspaces of a vector space V. Prove that W_{1} \cup W_{2} is a subspace of V iff W_{1}\subseteqW_{2} or W_{2} \subseteq W_{1}.

Homework Equations

See attempt at solution.

The Attempt at a Solution

My proof goes as such:

If W_{1}\subseteqW_{2} then the union of those subspaces is W_{2}, therefore, by the given, it the union is a subspace of V.
The same logic is used to argue the other subset.

I'm not sure if this is correct, and additionally, I'm not sure if its a logical proof. I feel it is a little cyclical maybe. Thanks for all the help, I'm having a tough time with this text.

 

[clamtrox]

Yeah that's half of the solution right there.

Then you need to also show (it's iff = if and only if), given that W_1 \cup W_2 is a subspace, either W_1 \subset W_2 or W_2 \subset W_1

 

[zapz]

Okay cool. Thanks. I'm self studying this book as a first exposure to linear algebra so I'm sure I'll be posting some more questions.