Coding Theory- minimum weight of dual codes

Question

Let C be a linear code in (F^n)q with minimum weight w(C) = 1. Prove that the dual code C^⊥ has no codewords of weight n.

I think that I might have to do a proof by contradiction but i'm not sure where to start, apart from beginning to prove that c^⊥ has a codeword of weight n.

Answer 1

Take a codeword $c\in C$ with minimum weight 1, say $c=\alpha e_i$, where $\alpha\ne0$ is a scalar and $e_i$ is the $i$th unit vector.

For each word $w=\sum_j \alpha_je_j$ with full weight $n$, the scalar product will give $\langle c,w\rangle = \alpha\alpha_i\ne 0$. So $w$ cannot lie in the dual code.