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## Homework Statement

V is a vector space over F.

Prove [tex]r\vec{0} = \vec{0}[/tex]

for all r in F and v in V.

## Homework Equations

Properties of a vector space:

1. associativity

2. commutativity

3. zero vector

4. additive inverse

5. scalar multiplication:

1u = u (and a few others)

I worked this similar to another proof:

[tex]r\vec{0} = r(\vec{0} + \vec{0})[/tex]

[tex]r\vec{0} = r\vec{0} + r\vec{0}[/tex]

[tex]r\vec{0} + (-r\vec{0}) = (r\vec{0} + r\vec{0}) + (-r\vec{0})[/tex]

[tex]r\vec{0} + (-r\vec{0}) = r\vec{0} + (r\vec{0} + (-r\vec{0}))[/tex]

[tex]\vec{0} = r\vec{0} + \vec{0}[/tex]

[tex]\vec{0} = r\vec{0}[/tex]

There may be other issues but my main one is that I assume:

[tex]r\vec{0} + (-r\vec{0}) = \vec{0}[/tex]

where in our definitions we only give(ie, not multiplied by a scalar):

[tex]\vec{v} + (-\vec{v}) = \vec{0}[/tex]

Now if I try to prove that rv + -rv = 0 I have to use the proof of the original problem !

Am I going about this the wrong way?