How can I visualize a "span" of a set of vectors?

Question

Can someone please help me visualize this concept, I know already what span of set of vectors is, but I am interested to know how it looks visually, thanks.

Answer 1

You could think of it as the set of all destination points you could arrive at by starting at the origin and successively moving any distances (forward or backward or none) in directions pointed in by that set of vectors. You get to pick which vectors are used and how far to move in those directions.

You can also use any of the vectors any number of times. Doesn't change anything -- nobody said the journey had to be "efficient". You could always optimize things (that's what you're doing when you reduce the spanning set to a basis for the span).

It doesn't matter if there are redundant vectors in the set. So in $\mathbb R^2$, the sets $\{(1,0),(0,1)\}$ and $\{(1,0),(0,1), (3,-7)\}$ have the same span. There may be various "journeys" that arrive at the same point.

Answer 2

Consider the subset of $\mathbb{R}^3$, $S = \{(1,0,0),(0,1,0)\}.$ The set $S$ is linearly independent and is a basis of $\mathbb{R}^2$. Thus, $S$ spans the $xy-$plane. So $\text{span}_{\mathbb{R}}\{(0,1,0),(1,0,0)\} = \mathbb{R}^2$. Other complicated combinations like $T=\{(1,2,3),(3,0,4)\}$ will be planes in $3D$.

As you go to higher dimensions, this may not be so easy to visualize, but these are some of the easiest examples to visualize.