Thread: 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 v for each , then u (u + v) for each .

Now I am thinking I can prove this using the contrapositive. I believe the contrapositive is: If u = (u+v) for some , then u = v for some .

If this is the correct negation then I know that u = (u+v) = u+ v so I need to find a way to make u+ v = 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

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 v for each , then u (u + v) for each .

Now I am thinking I can prove this using the contrapositive. I believe the contrapositive is: If u = (u+v) for some , then u = v for some .

If this is the correct negation then I know that u = (u+v) = u+ v so I need to find a way to make u+ v = 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:

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.