# Thread: Properties of Vector Spaces

1. ## Properties of Vector Spaces

Hello. I have just started doing proofs involving Linear Algebra and I need some guidance and a little push in the right direction. Here is the result I am trying to prove:
Let V be a vector space containing nonzero vectors u and v. Prove that if u $\neq$ $\alpha$v for each $\alpha \in \mathbb{R}$, then u $\neq$ $\beta$(u + v) for each $\beta \in \mathbb{R}$.

Now I am thinking I can prove this using the contrapositive. I believe the contrapositive is: If u = $\beta$(u+v) for some $\beta \in \mathbb{R}$, then u = $\alpha$v for some $\alpha \in \mathbb{R}$.

If this is the correct negation then I know that u = $\beta$(u+v) = $\beta$u+ $\beta$v so I need to find a way to make $\beta$u+ $\beta$v = $\alpha$v. Im not quite sure how to go about doing this so I am hoping someone here can push me in the right direction. Thanks

2. ## Re: Properties of Vector Spaces

Originally Posted by Twinsfan85
Hello. I have just started doing proofs involving Linear Algebra and I need some guidance and a little push in the right direction. Here is the result I am trying to prove:
Let V be a vector space containing nonzero vectors u and v. Prove that if u $\neq$ $\alpha$v for each $\alpha \in \mathbb{R}$, then u $\neq$ $\beta$(u + v) for each $\beta \in \mathbb{R}$.

Now I am thinking I can prove this using the contrapositive. I believe the contrapositive is: If u = $\beta$(u+v) for some $\beta \in \mathbb{R}$, then u = $\alpha$v for some $\alpha \in \mathbb{R}$.

If this is the correct negation then I know that u = $\beta$(u+v) = $\beta$u+ $\beta$v so I need to find a way to make $\beta$u+ $\beta$v = $\alpha$v. Im not quite sure how to go about doing this so I am hoping someone here can push me in the right direction. Thanks
I'lltake over here:

$u=\beta(u+v)~\implies~u=\beta u + \beta v ~\implies~u-\beta u = \beta v ~\implies~ \\ (1-\beta) u = \beta v ~\implies~ u = \underbrace{\frac{\beta}{1-\beta}}_{call \ this\ \alpha} \cdot v$

3. ## Re: Properties of Vector Spaces

one caveat: clearly β = 1 would be a "bad choice". so we need to know u = β(u + v) never happens when β = 1. which isn't too hard:

u = u + v implies v = 0, contradicting that v is, by stipulation, a non-zero vector.