A nowhere-zero point in linear mappings

We state the following conjecture and prove it for the case whereq is a proper prime power:Let A be a nonsingular n by n matrix over the finite field GFqq≧4, then there exists a vector x in (GFq)n such that both x and Ax have no zero component.

[1]  Noga Alon,et al.  Balancing sets of vectors , 1988, IEEE Trans. Inf. Theory.

[2]  Alexander Schrijver,et al.  The Blocking Number of an Affine Space , 1978, J. Comb. Theory, Ser. A.