# Linear Algebra - Vector Spaces

1. Sep 28, 2005

### 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.

2. Sep 28, 2005

### 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.

3. Sep 28, 2005

### Delta-One

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

4. Sep 28, 2005

### HallsofIvy

Staff Emeritus
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?

5. Sep 28, 2005

### 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?

6. Sep 29, 2005