Linear Algebra
2.810 min read

Span: The Reach of a Set of Vectors

The span of a set of vectors is the collection of all possible linear combinations you can form from them. It answers: "What can we reach?"

A single nonzero vector a\mathbf{a} spans a line through the origin — all scalar multiples cac\mathbf{a}. Two vectors a1,a2\mathbf{a}_1, \mathbf{a}_2 that point in genuinely different directions span a plane through the origin. If they point in the same direction, they still span only a line.

The span always contains the zero vector (take all coefficients to be zero) and is closed under addition and scalar multiplication — making it a subspace.

Formal View

Definition 2.8 — Span
The span of vectors a1,…,ak∈Rm\mathbf{a}_1, \ldots, \mathbf{a}_k \in \mathbb{R}^m is
Span({a1,…,ak})={c1a1+⋯+ckak:ci∈R}\text{Span}(\{\mathbf{a}_1, \ldots, \mathbf{a}_k\}) = \{c_1\mathbf{a}_1 + \cdots + c_k\mathbf{a}_k : c_i \in \mathbb{R}\}
By convention, Span(∅)={0}\text{Span}(\emptyset) = \{\mathbf{0}\}.

Interactive Visualization

Span Visualizer

Why This Matters

The span of the columns of a matrix is exactly the set of outputs the matrix can produce — a fundamental concept for solvability.

  • A system Ax=bA\mathbf{x} = \mathbf{b} is solvable iff b∈Span\mathbf{b} \in \text{Span}(columns of AA)
  • The span of sensor measurements tells us what signals can be detected
  • In graphics, the span of basis colors determines what colors can be rendered

Quiz

Question 1

The span of a single nonzero vector in R3\mathbb{R}^3 is:

Common Mistakes

  • Thinking span only refers to integer multiples — it includes all real scalar multiples.
  • Assuming more vectors always gives a larger span — adding a vector already in the span changes nothing.