# Linear Algebra Done Right Exercise 1 B Q5

In the book Linear Algebra Done Right Exercise 1.B Q5, the question asked to show that in the definition of a vector space (1.19), the additive inverse condition can be replaced with the condition that $$0v = 0 \quad\text{for all}\quad v\in V$$

But I don't quite understand what is the meaning of additive inverse condition. What is the original "condition" in the definition of vector space (1.19) where it said $$\text{for every}\quad v\in V, \text{there exists}\quad w\in W \quad\text{such that}\quad v + w = 0$$;

And how, after replacing the original condition with the condition that $$0v = 0 \quad\text{for all}\quad v\in V$$

All you have to do is prove that the sentence $$0v=0\text{ for all v\in V}$$ implies the sentence $$\text{for all v\in V, there exists w\in V such that } v+w=0,$$ which is what is shown in the answer at your link.
An inverse $$w$$ of an element $$v$$ usually describes a special element in an algebraic structure, in this case a vector space, that combines using some operator $$w \circ v$$ to give the identity. In particular, for our example, this operator is $$w + v$$ and the identity of addition is 0.
The condition $$\text{for every}\quad v\in V, \text{there exists}\quad w\in W \quad\text{such that}\quad v + w = 0$$ is named the additive inverse condition because for any vector $$v$$, we can find an element $$w$$ such that $$v + w = 0$$. In this case, $$w$$ is the additive inverse.