rom the axioms of a vector space, prove that −(v + w) = (−v) + (−w), for all vectors v, w

Sure, here is the proof:

1. We know that for any vector v, v + (-v) = 0. This is the definition of additive inverse in a vector space.

2. Now, consider the vector v + w. Its additive inverse is -(v + w), which means (v + w) + (-(v + w)) = 0.

3. We also know that vector addition is associative, so we can write this as v + (w + (-(v + w))) = 0.

4. Now, we can use the definition of additive inverse again to replace w + (-(v + w)) with -v. This gives us v + (-v) = 0, which we know is true.

5. Therefore, -(v + w) = -v + -w.

This proof relies on the axioms of a vector space, specifically the existence of an additive inverse for every vector and the associativity of vector addition.