the span of a given set of vectors is the collection of all the possible linear combinations of the vectors of the set, i.e.:
let be a vector space, let and , then
let , then
let
we have proven that is a subset of , now with that out of the way we need to check the 3 conditions that need to be satisfied for a subset to be a subspace
condition 1: condition 2: addition closure
let so there exist so that:
condition 3: multiplication closure
let so there exist , such that
by definition of span, a span of a set of vectors is the collection of all the possible linear combinations using said vectors
the span of a single vector would just be its own line expanded across the entire 3d region, because we cant reach other dimensions by constant multiplication or by addition of the vector to itself
the span of 2 linearly independant vectors is however more interesting because it would be visualized as a grid across the 3d space
for simplicity, we take the relatively simple vectors that lie on the grid which covers the and axes