TITLE: What is the difference between linearly and affinely independent vectors?
QUESTION [7 upvotes]: What is the difference between linearly and affinely independent vectors? Why does affine independence not imply linear independence necessarily? Can someone explain using an example?

REPLY [2 votes]: The set $\{ v_1,...,v_n\}$ is linearly independent if for $c_1, c_2,...,c_n$, not all zero, such that $$\sum_{k=1}^n c_kv_k =0 \Leftrightarrow c_j=0, \: \forall j=1,2,...,n.$$ The set is called affinely linearly independent if they are linearly independent and also $$ \sum_{k=1}^n c_k=1.$$