$v_0,v_1,...,v_k$ are affinely independent iff $v_1-v_0, v_2-v_0,...,v_k-v_0$ is linearly independent.
If I were given a finite set of (affinely independent) vectors with numerical coefficients, for example, I could easily rename the vectors (choose any vector in the set to be $v_0$) and show affine independence. But I have no idea how to do this for an arbitrary set of vectors with arbitrary coefficients.
You could note the the matrix transforming $v_1-v_0,v_2-v_0,\ldots,v_n-v_0$ to say $v_0-v_1,v_2-v_1,\ldots,v_n-v_1$ is non-singular.
Alternatively you could give a more symmetric condition for affine (in)dependence, for instance, $v_0,\ldots v_n$ is affinely independent iff the only $(\lambda_0,\lambda_1,\ldots,\lambda_n)$ with both $\sum\lambda_i=0$ and $\sum\lambda_iv_i=0$ is $(0,0,\ldots,0)$.