Span

Precise definition

The span of vectors $v_1,\ldots,v_k$ is the set of all linear combinations $\operatorname{span}\{v_i\}=\{c_1v_1+\cdots+c_kv_k:c_i\in\mathbb F\}$. It is the smallest subspace containing every listed vector.

Notation and mathematical language

To test whether $b$ lies in the span of matrix columns, solve $Ac=b$. Consistency gives coefficients; inconsistency proves nonmembership. A spanning list may be dependent and nonunique, while a basis is a minimal independent spanning list.

Conceptual picture

Varying coefficients sweeps every attainable direction. Adding a vector already in the span changes the generating list but not the subspace. Span describes reachability, not just a picture of the listed arrows.

Conditions and key results

All coefficients range over the stated field. Affine combinations whose coefficients sum to one define affine hulls, not ordinary spans. The span always contains zero, so a line or plane not through zero cannot be a linear span by itself.

A reliable strategy

1. Put generators in columns and the target as an augmented column.
2. Row-reduce to test consistency and solve for coefficients.
3. For the full spanned subspace, keep original pivot columns as a basis and state a parametric description if useful.
4. Verify a membership result by reconstructing the target or an exclusion by the contradiction row.

Fully worked example

Interpretation and application

Span identifies model spaces, controllable states, feature spaces, and solution families. A target near but outside a span motivates projection and least squares rather than declaring exact membership.

Common mistakes

Verification and reasonableness

• Reconstruct a target from solved coefficients.
• Check pivot columns and dimension against the geometric description.
• Test closure and the presence of zero for a claimed spanned set.

Practice

1. Does $(3,6)$ lie in $\operatorname{span}\{(1,2)\}$?
2. What is the span of the empty set?
3. Can a dependent list still span a space?

Further deduction

The column space of a product satisfies $\operatorname{Col}(AB)\subseteq\operatorname{Col}(A)$ because every column of $AB$ is a combination of columns of $A$. Equality can fail when $B$ does not generate enough coefficient directions. This observation leads directly to the rank inequality $\operatorname{rank}(AB)\le\operatorname{rank}(A)$.

A span can be invariant under a transformation. If $T(v_i)$ lies in $\operatorname{span}\{v_1,\ldots,v_k\}$ for every generator, linearity guarantees $T$ maps the entire span into itself. It is enough to test generators, a powerful simplification in eigenvalue, recurrence, and dynamical problems.

For polynomials of degree at most two, coordinate vectors relative to $\{1,x,x^2\}$ turn span questions into ordinary column-space questions. For example, $1+x$, $x+x^2$, and $1-x^2$ have coefficient columns $(1,1,0)$, $(0,1,1)$, and $(1,0,-1)$; their dependence can be tested by row reduction. This coordinate translation is justified because equality of polynomials is coefficient-wise equality, not merely agreement at one input.

Related topics

• Linear Independence
• Basis
• Subspaces