If $u_i$ are affinely independent, are they also linearly independent?

The reverse implication is not true, consider the following three vectors in $\mathbb{R}^3$: $u_0=e_1, u_1=2e_1$ and $u_2=e_1+e_2$. The three vectors are affinely independent but not linearly independent.

The best way to understand the difference is from the picture on the wikipedia page. Think of $u_0$ as the "base vertex"and all the other $u_i$ as the positions of the other vertices. $u_i-u_0$ is then the position vector of all the other vertices relative to the base vertex, and we need these to be linearly independent (so that $u_0,\dots, u_k$ are affinely independent) so that we don't end up with three collinear vertices which would mess up our idea of what a simplex should be.

It should also be noted that the point $u_0$ is not special in the definition of affine indepenence, but in fact if $u_0,\dots,u_k$ are affinely independent then for anyfor any $j$, $u_0-u_j,\dots,u_k-u_j$ are all linearly independent. Check out this wikipedia page: http://en.wikipedia.org/wiki/Affine_space#Affine_combinations_and_affine_dependence