There is a theorem that says: if $M=\{v_1,v_2,...,v_m\}$ are a set of vectors such that they are affine independent (i.e., $\{v_2-v_1,v_3-v_1,...,v_m-v_1\}$ are linearly independent), then for any vector $v$ that is in the convex hull of $M$, it can be written as a unique convex combination of the vectors in $M$.
My first question is: here the unique representation means $v=\sum_{j=1}^m K_jv_j$, where $\sum_{j=1}^mK_j=1$ and $K_j\geq 0,~\forall~1\leq j\leq m$. In particular, some of those $K_j$'s could be zero right?
My second question is: I couldn't feel the significance and importance of this theorem. Isn't that obvious? I mean, if we have a set of vectors, then we construct the convex hull of this set of vectors, which is taking all possible convex combinations of this set of vectors, then if you pick an arbitrary vector from this generated convex hull, it is for sure should be able to written as a unique representation of those vectors in the set that generates this convex hull. I guess my question is: what's the role played by this condition "$M$ is a set of affine independent vectors"? What if my set $M$ is not a set of affine independent vectors, then after I generate the convex hull (I can still do that by taking all possible convex combinations of the vectors in $M$), then if I pick an arbitrary vector from this convex hull, then why I cannot uniquely express this vector as a convex combination as those vectors in set $M$?
Use the analogy to linear spaces:
If $b_1,...,b_n$ are linearly independent then $x \in \operatorname{sp} \{ b_k \}$ iff there are unique $\lambda_k$ such that $x =\sum_k \lambda_k b_k$.
Note that some of the $\lambda_k$ may be zero or negative.
Similarly, if $a_1,...,a_n$ are affinely independent then $x \in \operatorname{aff} \{ a_k \}$ iff there are unique $\mu_k$ such that $x =\sum_k \mu_k a_k$ and $\sum_k \mu_k = 1$.
Note that some (but not all) of the $\mu_k$ may be zero or negative.
In the latter case, if $x =\sum_k \mu_k a_k$, with $\sum_k \mu_k = 1$, then $x \in \operatorname{co} \{ a_k \}$ iff $\mu_k \ge 0$.
Notes:
A few related fact may help: (i) $\{ a_k \}_k$ are affinely independent iff $\{ \binom{1}{a_k} \}_k$ are linearly independent. (ii) $\{ a_k \}_k$ are affinely independent iff for any $k_0$ the collection $\{ a_k -a_{k_0} \}_{k \neq k_0} $are linearly independent