# Proof involving subsets of a vector space

1. Jul 18, 2012

### zapz

1. The problem statement, all variables and given/known data

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}$$\subseteq$$W_{2}$ or $W_{2}$ $\subseteq$ $W_{1}$.

2. Relevant equations

See attempt at solution.

3. The attempt at a solution

My proof goes as such:

If $W_{1}$$\subseteq$$W_{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.

2. Jul 18, 2012

### 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 $W_2 \subset W_1$

3. Jul 18, 2012

### 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.