TITLE: Why do we need "span" in linear algebra?
QUESTION [11 upvotes]: In my linear algebra course in university we started learning about span and I was curious what is it good for? and if someone know, how does it relate to 3D graphics?
Thank you.

REPLY [4 votes]: Linear algebra is not a theory about vectors, it's a theory about spaces.
In set theory, you can reason about a set $S$ by representing elements of it symbolically, i.e. with variables $x, y, z,$ whathaveyou. If we're reasoning about three elements at a time, we can think of this as probing $S$ with a function from a three-element set $\{ x,y,z \}$ to $S$ (i.e., an arrow $\{ x, y, z \} \rightarrow S$).
In linear algebra, analogously, you use the free vector space $F(\{ x,y,z \})$ over $\{ x,y,z \}$, which is like the standard 3D space over the field of scalars, to probe a vector space V, by way of a linear map $F(\{ x,y,z \}) \rightarrow V$. Instead of getting a subset of atmost three elements of $S$, you get a subspace of atmost dimension 3 of V. This is the image of the linear map, and is the same as the span of the values of $x, y,$ and $z$ under the map.
The "dimension at most 3" part is relevant for linear maps in graphics. If you want to make sure you project the outline of a cube into 2D so that you can see every face, what you're saying is you want no two edges drawn to overlap in a line segment. So, for every point of the cube, you want it so that if you take that point as your origin for a 2D vector space, every two points connected to it by an edge span the plane. Embedding the 2D space into 3D as a plane offset from the origin, you can use affine span instead of linear span and a reference to each origin: Try taking the affine span of (1,0) and (0,1) in 2D, and then its convex hull. All three of these spans are closure operators, and they are generally helpful for solving and problems to do with arranging things a certain way.
But something I've just had to draw is this 3D solid which has a special point at the center and spokes coming out of it, and where the points at the end of the spokes are all distributed around the unit sphere around the special point. Then, since you can see all the points from the center, you only need to check that if you take the center to be the origin, you can distinguish all the points as you see them on the sphere from the origin. Hence, to have a 3D projection like this that keeps all the points and edges separate, it suffices for the linear span of every two spoke terminals to be 2D and not 1D. Why say it that way? Because you have to sometimes look at the shape before you draw it, and if you can't tell from the drawing whether the points are the same, you need a way to test for that. So for this problem, the linear span is a tool for designing a projection, and describing a constraint on the program in a way that the same libraries you're probably writing the project code in can deal with.