Kunze Linear Algebra · Problem 3.5.14

Linear Transformations

Mathematics — Methods · separating-functional

Problem

Let FF be a field of characteristic zero and let VV be a finite-dimensional vector space over FF. If α1,…,αm\alpha_1, \ldots, \alpha_m are finitely many vectors in VV, each different from the zero vector, prove that there is a linear functional ff on VV such that f(αi)≠0,i=1,…,m.f(\alpha_i) \neq 0, \quad i = 1, \ldots, m.
Intuitive explanation

The key insight is that you're trying to avoid finitely many "bad" hyperplanes simultaneously. Each vector αi\alpha_i defines a hyperplane (the kernel of any functional that annihilates it), and you need to find just one linear functional that misses all of them. Since VV is finite-dimensional, the union of finitely many hyperplanes cannot cover the entire dual space V∗V^*—so a functional outside this union must exist.

Intuitive explanation

The key insight is that you're trying to avoid finitely many "bad" hyperplanes simultaneously. Each vector αi\alpha_i defines a hyperplane (the kernel of any functional that annihilates it), and you need to find just one linear functional that misses all of them. Since VV is finite-dimensional, the union of finitely many hyperplanes cannot cover the entire dual space V∗V^*—so a functional outside this union must exist.

Browse the problem library