# Showing that "zero vector space" is a vector space

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1. Oct 9, 2016

### Mr Davis 97

1. The problem statement, all variables and given/known data
Let $\mathbb{V} = \{0 \}$ consist of a single vector $0$ and define $0 + 0 = 0$ and $c0 = 0$ for each scalar in $\mathbb{F}$. Prove that $\mathbb{V}$ is a vector space.

2. Relevant equations

3. The attempt at a solution

Proving that the first six axioms of a vector space are true is trivial, I am just on the distributive axioms.

So if I want to show that $a(x + y) = ax + ay$ is true, where a is a scalar and x and y are vectors, is it sufficient to make the argument that $a(0 + 0) = a0 = 0 = a0 + a0$? Does this show that a distributes over the vector $0$?

2. Oct 9, 2016

### Simon Bridge

I guess ... since the space only contains one vector you can also show that the last two are equivalent to the previous ones.
ie. u=v, then a(u+v)=a(u+u)=2au = scalar multiplication axiom already tested.
ie u=v, then RHS: a(u+v)=2au and LHS=au+av = 2au so RHS=LHS

3. Oct 9, 2016

### PeroK

Yes, that's exactly how to do it.