I'm not familiar with Affine combination, but I came across this concept in a paper and tried to make sure whether I understand it correctly.
Suppose in a D-dimensional vector space we have three linearly independent vector $v_1,v_2,v_3$. The linear combination of this three vector spans a 3-dimensional subspace. Is that right to say that the affine combination of the three vectors spans a 2- dimensional subspace? Furthermore, the convex combination will just be a finite area of the span of affine combination?
I may well mess up a lot of concepts here. Please feel free to point them out. Thank you
Yes, you're right that the affine span of three (affinely independent) vectors will be an affine 2-dimensional plane. Note that vectors may be linearly dependent and still be affinely independent. The relevant definition is this:
$v_0,\dots,v_k$ are affinely independent if the only solution of $c_0v_0+c_1v_1+\dots+c_kv_k=0$ with $c_0+c_1+\dots+c_k=0$ is $c_0=c_1=\dots=c_k=0$.
You can check that $v_0,\dots,v_k$ are affinely independent if and only if the $k$ vectors $v_1-v_0$, $\dots$, $v_k-v_0$ are linearly independent (and this definition does not depend on your choice of which one is $v_0$).
The next thing to check is that the affine span of $v_0,\dots,v_k$ (i.e., the affine subspace they span) consists precisely of all vectors $$x=c_0v_0+c_1v_1+\dots+c_kv_k \quad\text{with}\quad c_0+c_1+\dots+c_k=1.$$ (Hint: Write such an $x$ as $v_0+\lambda_1(v_1-v_0)+\dots+\lambda_k(v_k-v_0)$ for some scalars $\lambda_1,\dots,\lambda_k$.) It's worth thinking about what subset you get in the affine span when you require, for example, all $c_i\ge 0$. (Can you interpret, then, what it means to have one particular $c_i\ge 0$?)