Showing that "zero vector space" is a vector space

[Mr Davis 97]

Homework Statement

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.

Homework Equations

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 $0$?

 

[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

 

[PeroK]

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 $0$?

Yes, that's exactly how to do it.