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.

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 can rearrange this equation to get -(v + w) = 0 - (v + w).

4. We also know that for any vector v, 0 - v = -v. This is because 0 + (-v) = -v, and addition is commutative in a vector space.

5. Therefore, we can rewrite the right side of the equation from step 3 as -(v + w) = -v - w.

6. Finally, we know that for any vectors v and w, -v - w = (-v) + (-w). This is because subtraction is the same as adding the additive inverse.

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

So, we have proved that for all vectors v and w in a vector space, -(v + w) = (-v) + (-w).