Why do we need "span" in linear algebra?

Given a set of vectors, what can you do with them? Well, by the axioms of a vector space, you can add and subtract, or multiply by a scalar -- and this is exactly what the span of a set of vectors is! You're given a list of vectors, and told "Here you go! You can only play with these vectors. See what you can make with them." The set of all things you can make is the span of those vectors.

That the span is a subspace is nice -- it's always good to have objects that are closed under certain operations, and subspaces are just that: closed under vector addition and scalar multiplication. Adding or subtracting linear combinations, or multiplying them by a scalar, is yet another linear combination. This isn't true for most generic sets of vectors, but definitely true for the span of a set of vectors. So, spans generally behave in a nice way, nicer than the set of vectors you started with.

The span of a set of vectors is a the subspace generated by these vectors. This is an inherently useful object. This has also immediate geometric meaning.

• The span of a vector $v$ is exactly the line through the origin in direction $v$.

• For two vectors $v,w$, not co-linear, the span is the plane containing the origin "spanned" by $v$ and $w$, that is containing the line in direction $v$ and the line in direction $w$.

"Span" can be thought of as a way to formalize the concept of a coordinate system.

There are many situations in geometry, physics, etc. where one needs to set up a coordinate system determined by two vectors $\vec v,\vec w$ that play the role of the unit $x$ and $y$-vectors.

Now ask yourself: what is the relation between the two vectors $\vec v,\vec w$ and the set of vectors in the coordinate system?

Answer: the set of vectors in the coordinate system is the span of $\vec v$ and $\vec w$.

Why is this true? Draw a picture of the coordinate system, with origin $O$ where the $\vec v$ and $\vec w$ axes cross each other; choose a coordinate $s$ on the axis in the $\vec v$ direction; choose a coordinate $t$ on the axis in the $\vec w$ direction; draw the usual picture that we learn in precalculus for locating the point $P(s,t)$ in this coordinate system. And then observe: the vector $OP$ is precisely the linear combination $$s \vec v + t \vec w $$ As you vary the choice of $s,t$, you get $$\text{Span}(\vec v,\vec w) = \{s \vec v + t \vec w \, \bigm| \, s,t \in \mathbb{R}\} $$