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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 $$0v = 0 \quad\text{for all}\quad v\in V$$

I can find the answer in this link.

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 $0v = 0 \quad\text{for all}\quad v\in V$

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The condition is the sentence that you've quoted.

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.

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I'm going to address your question of why it is called the additive inverse condition.

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.

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