Span and basis — Linear Algebra

You just met the two outcomes. Some pairs of vectors reach the whole plane; others only reach a line. The set of everything reachable deserves a name.

The span of a set of vectors is every linear combination of them — every point you can hit by scaling and adding.

Reading spans off examples

$\text{span}{(1,0), (0,1)}$ is the whole plane. Any $(a,b)$ is $a(1,0) + b(0,1)$ .

$\text{span}{(1,2)}$ is a line: all multiples of $(1,2)$ , forever in both directions.

$\text{span}{(1,2), (2,4)}$ is also that same line, because the second vector is $2 \times$ the first and contributes nothing new.

$\text{span}{(1,2), (0,1)}$ is the whole plane again. These two point in genuinely different directions, and together they get everywhere.

Every span contains the origin, always. Scale everything by zero and you land there. This is not a technicality; it is why every structure in this subject is anchored at the origin rather than floating loose.

A basis is a good set of ingredients

You want a set that reaches everything you care about, with nothing wasted. That is a basis: a set of vectors that spans the space and has no redundancy.

${(1,0), (0,1)}$ is the obvious basis for the plane, called the standard basis and written $\hat{\imath}, \hat{\jmath}$ .

But ${(1,1), (1,-1)}$ is equally valid. It also spans the plane with nothing wasted. It just measures the plane along diagonal directions instead.

The idea this sets up

There is no single correct basis. There are infinitely many, and every one of them assigns different coordinates to the same unmoved arrow.

Choosing a basis is choosing a language for describing space. Some languages make a problem look like a mess; others make the same problem trivial. Learning to switch is the payoff waiting at the end of this path.

First, though: how do you tell redundancy from genuine new direction?