Linear independence of orthogonal vectors.

Let {$v_1...v_k$} be a linearly independent subset of a vector space V. Let {$u_1...u_m$} be a linearly independent subset of a vector space $V_{perpendicular}$. (So for example, $v \cdot x = 0)$. I want to prove now that the whole list {$v_1...v_k, u_1, ...u_m$} is linearly independent.

How would I go about doing this? Thanks!