Linear indpendence of charactersPosted on May 18, 2019
Theorem. Let be an abelian group, let be a field, and suppose that are distinct characters of with values in , i.e., is group homomorphism for each . Then the characters are linearly independent, i.e., if such that for all , then it must be that .
If observe that since in particular for all . Thus implies .
Suppose that for some all sets of fewer than characters of are linearly indpendent. Let be distinct characters and such that for all . Let be distinct. Observe that since the characters are distinct there exists such that . Observe that by our choice of we have and for all . Therefore subtracting the two we see that for all . Since the term vanishes, by the inductive hypothesis we must have all the coefficients of the linear combination equal to zero. That is, we must have for all . In particular, . But by our choice of we have . Thus .
The above proof is inspired by the proof given in (1, Theorem 2.1).