Understanding Linear Algebra's Span Concept

[Eiskrele]

Homework Statement

This is a general question: What is span in linear algebra?

Homework Equations

span = linear distribution
span(v1, v2, ..., vk) = every linear combination of v1, v2, ..., vk

The Attempt at a Solution

I've try to visualize what span is, but I just can't "see" it. I don't understand what span is or what it does.

 

[Hurkyl]

Algebraically... you have an operation "multiply by a scalar" and an operation "add". The span of a set is contains everything you can produce by applying these operations to the vectors in the set.

 

[Eiskrele]

Ok. But that is the problem for me. Because then span must be an infinity large area. How can I visualize that? Like a ball? And what can I use span for?

 

[Mark44]

Ok. But that is the problem for me. Because then span must be an infinity large area. How can I visualize that? Like a ball? And what can I use span for?

Let's take it in steps. The real line can be thought of as a vector space of dimension one. The vector x that extends from the origin to 1 is one unit long and spans this space because every vector is some multiple of x.

Now consider the real plane, a vector space of dimension two. The vectors (1, 0) and (0, 1) span this space, because every vector in this space can be written as the sum of scalar multiples of these two vectors. For example, the vector (2, -8) = 2(1, 0) + (-8)(0, 1). The vectors (1, 0) and (0, 1) are not the only vectors that span the real plane; any two vectors that have different directions will span the plane.

 

[Hurkyl]

How can I visualize that?

Not everything needs to be "visualized" geometrically. However, a span is a vector space -- furthermore, it's a vector space contained in another vector space.

Are there any vector spaces at all you know how to visualize? Well, you've hopefully learned something about coordinates or isomorphisms or something -- you can use the vector space you already know how to visualize as your picture for the vector space you want to study.

And what can I use span for?

I know I'm going to sound silly -- but one stereotypical use is when you are interested the linear combinations of some vectors. Another common use is when you're doing a problem that you can convert into the question about linear combinations of vectors.

You know Gaussian elimination, right? And (reduced) row echelon form?

Both of those algorithms have to do with spans -- they are algorithms for simplifying the presentation of a span. In some sense, Gaussian elimination works by searching through the span of your equations, looking for ones that are easier to solve.