How Do I Prove Basic Vector Space Properties?

[Delta-One]

Hi,
I'm having trouble with these homework questions.

I have to prove that B*0v = 0v , where B is a scalar.

Also, I have to prove that if aX = 0v , then either a = 0 or X = 0
---where a is a scalar and X is a vector.


I know that I have to use the 8 axioms but I'm not sure where to begin.

 

[willworkforfood]

One of the axioms is that anything times a 0 vector is 0, unless that is the one you are trying to prove.

The zero vector will contain all zero components, therefore any scalar multiplied by 0 will also be 0 as 0*a is always zero for any scalar. For your first question, the proof depends on what "B" represents.

 

[Delta-One]

Thanks for answering the first question, but any ideas about the second (aX = 0)? These vector spaces really have me confused.

 

[HallsofIvy]

No, "0 times anything is 0" is never an axiom- it's too easy to prove!
And I don't like the idea of using components to prove this- not general enough.

What is B*(u+ 0v)? (u is a vector, 0v is the 0 vector)
What is u+ 0v?

If a is not 0, what is (aX)/a? If aX= 0 what does the answer to my question tell you?

 

[Delta-One]

I'm really sorry but I still don't see how to show that
B*0v = 0v
--where B is a constant.

Are you suggesting that I add the vector u to each side of the eqn?

 

[lurflurf]

I'm really sorry but I still don't see how to show that
B*0v = 0v
--where B is a constant.

Are you suggesting that I add the vector u to each side of the eqn?

what are your axioms?
One possible set includes
(for a, b scalars and v a vector)
(ab)v=a(bv)

to show for some vector v
v=0
show that for any vector u
u+v=v+u=u