Span and Linear Combinations

Everything You Can Reach

A linear combination of vectors $v_1, \dots, v_k$ is any expression $$c_1 v_1 + c_2 v_2 + \dots + c_k v_k$$ with scalar coefficients $c_i$. It is the most basic way to build new vectors from old ones — adding scaled copies. The collection of all linear combinations of a given set is its span, written $\operatorname{span}\{v_1, \dots, v_k\}$ — the entire region those vectors can reach.

The span is always a subspace (it contains $\mathbf{0}$, taking all $c_i = 0$, and is closed under the operations). It is, in fact, the smallest subspace containing the given vectors. Geometrically:

• The span of a single nonzero vector is the line through it.
• The span of two non-parallel vectors in the plane is the whole plane.
• The span of two parallel vectors is just a line (the second adds nothing new).

A set of vectors is a spanning set for a space $V$ if $\operatorname{span}$ of the set is all of $V$ — every vector in $V$ is some linear combination of them. For example, $(1, 0)$ and $(0, 1)$ span $\mathbb{R}^2$: any $(a, b) = a(1,0) + b(0,1)$.

Determining whether a vector $w$ is in the span of others is exactly solving a linear system: $w = c_1 v_1 + \dots + c_k v_k$ is a system in the unknowns $c_i$, which has a solution precisely when $w$ lies in the span. This ties spanning directly to the Gaussian-elimination machinery of Unit 3.

Common pitfall: assuming that more vectors always span more. Adding a vector that is already a linear combination of the others does not enlarge the span — two parallel vectors span only a line, not a plane. What grows the span is a vector pointing in a genuinely new direction (one not already reachable). Redundant vectors expand the list without expanding the reach.