Abstract Algebra: Theory and Applications · Problem 20.10

Fields — Thomas Judson

Mathematics — Abstract Algebra · vector-space-properties

Problem

Let VV be a vector space over FF. Prove that −(αv)=(−α)v=α(−v)-(\alpha v) = (-\alpha)v = \alpha(-v) for all α∈F\alpha \in F and all v∈Vv \in V. Hint. Since 0=α0=α(−v+v)=α(−v)+αv0 = \alpha \mathbf{0} = \alpha(-v + v) = \alpha(-v) + \alpha v, it follows that −αv=α(−v)-\alpha v = \alpha(-v).
Intuitive explanation

The core concept here is that scalar multiplication and vector negation interact predictably because they're both defined through the same underlying axioms of a vector space. The key insight is to use the distributive property of scalar multiplication over vector addition, along with the fact that additive inverses are unique: if two things sum to zero, one must be the negative of the other. The hint shows how to establish one equality; the other follows by a similar argument using the distributive property in the scalar component instead.

Intuitive explanation

The core concept here is that scalar multiplication and vector negation interact predictably because they're both defined through the same underlying axioms of a vector space. The key insight is to use the distributive property of scalar multiplication over vector addition, along with the fact that additive inverses are unique: if two things sum to zero, one must be the negative of the other. The hint shows how to establish one equality; the other follows by a similar argument using the distributive property in the scalar component instead.

Browse the problem library