What is the definition of affine independence then?

Question

The following is the definition of affine dependence (This definition is from James Oxley book, second edition "matroid theory")

Definition of affine dependence:

A multiset $\{ \underline{v_1},\dots, \underline{v_k} \},$ the members of which are in $V(m, \mathbb F)$ ( the vector space of dim. m over the field $\mathbb F$), is affinely dependent if $k \geq 1$ and there are elements $a_1, a_2, \dots ,a_k$ of $\mathbb F$ that are not all zero such that $\sum_{i=1}^{k}a_i v_i = 0$ and $\sum_{i=1}^{k}a_i = 0.$ A multiset of elements from $V(m, \mathbb F)$ is affinely independent if it is not affinely dependent.

My question is:

What is the definition of affine independence then? How it deals with this condition $\sum_{i=1}^{k}a_i = 0.$ Could someone explain this to me please?

Answer 1

The condition is that the only solution of $\sum a_iv_i=0$ and $\sum a_i=0$ is the trivial one $a_i=0$, all $i$.

You can check that $v_1,\dots,v_k$ form an affinely (in)dependent set if and only if $v_2-v_1,\dots,v_k-v_1$ form a linearly (in)dependent set. The condition holds subtracting any $v_j$ rather than $v_1$.