Constructing vectors in general position Let a real $k\times n$  ($k\le n$) matrix ${\bf A}$ with the property that any collection of $k$ columns is full rank. 

Q: Is there an efficient way to deterministically find a vector ${\bf a}$ such that the augmented matrix ${\bf A}&#39; = [{\bf A}\;{\bf a}]$ preserves the same property as ${\bf A}$: any $k$ columns are full rank.

Relevant Sidenote: A matrix that has this property is the generator of an $(n,k)$ Reed-Solomon Code: adding columns that preserve its Vandermonde structure preserves the rank property.
If you choose $\mathbf{a}$ uniformly at random from the hypercube $[0,1]^n$, the matrix $[\mathbf{A}~ \mathbf{a}]$ will have the desired property with probability $1$.