Graphing linear, affine, and convex combinations For the vectors (2, 1) and (1, 3), how would I graph each of the three combinations?
Here are my thoughts (sorry might be totally wrong):


*

*linear - plane connecting the two points

*affine - infinite line connecting the two points

*convex - triangle with vertices at origin and the two points

 A: You're actually right:
1) Linear combinations of $v_1 = (2,1)$ and $v_2=(1,3)$ spans the entire plane, since they form a basis of this two dimensional vector space.
2) Affine combinations of $v_1$ and $v_2$ are the set $A$ of points parametrically described by the real scalar $\alpha$ such that if  $u$ $\in$ $A$ then $$u=\alpha v_1+(1-\alpha)v_2=v_2+\alpha(v_1-v_2)$$ Now obviously this represents a one-dimensional vector space, with sum of two vectors defined as the vector whose $\alpha$ is the sum of the respective parameters of the two vectors. From this and the fact that $v_1$ and $v_2$ belong to $A$ we conclude that the affine combination is in fact the line connecting the heads of both vectors if their tails are positioned at a common point (the origin, for instance).
3) The convex combination is a truncated version of the affine combination of the vectors, such that only values of $\alpha$ in $[0,1]$ are allowed. This excludes both portions of the affine line outside the central region.
