Question

Identify which vector space axiom justifies each step in the following proof. For lines n=6 through 10 , justify why we can go from line n−1 to line n. Theorem: The additive inverse of every vector is unique.

1. Let V be a vector space, and let v be an element of V.

Answer 1

The vector space axiom that justifies going from line n−1 to line n in this proof is the axiom of closure under vector addition.

Step-by-step explanation:

The vector space axiom that justifies going from line n−1 to line n in this proof is the axiom of closure under vector addition. This axiom states that if u and v are vectors in a vector space V, then the sum u + v is also a vector in V. In this case, the proof is using the fact that if v is an element of V, then v + (-v) = 0, where 0 is the zero vector or additive identity in the vector space. So, when we go from line n−1 to line n, we are adding v to its additive inverse -v, resulting in the zero vector 0.