The span of a set of vectors is the collection of all their linear combinations — every vector that can be built by adding scaled copies of the given vectors. It is always a subspace, and it is the smallest subspace containing the original set. Span is the other half of what makes a basis: the half that guarantees complete coverage.
Definition
Given vectors v1,v2,…,vk in a vector space V, their span is the set of all linear combinations:
Every choice of scalars c1,…,ck produces one vector in the span. As the scalars range over all real numbers, the span sweeps out an entire subspace of V.
The span is always a subspace — the smallest subspace of V that contains all of v1,…,vk. Adding two linear combinations of these vectors produces another linear combination, and scaling a linear combination by a scalar produces another, so both closure conditions hold automatically.
By convention, the span of the empty set is {0}. With no vectors to combine, the only reachable point is the zero vector (the combination with no terms).
Geometric Interpretation
In R2 and R3, the span of a set of vectors is a flat subspace whose shape depends on how many independent vectors the set contains.
The span of a single nonzero vector v is the line {tv:t∈R} — a one-dimensional subspace passing through the origin in the direction of v.
The span of two non-parallel vectors is a plane through the origin — a two-dimensional subspace. If the two vectors happen to be parallel, one is a scalar multiple of the other, and the span collapses to a line.
The span of three vectors in R3 that do not all lie in a single plane is all of R3. If the three vectors are coplanar, the span is a plane. If two of the three are parallel, the span may be only a line.
The pattern is consistent: the dimension of the span equals the number of independent vectors in the set, regardless of how many total vectors are present. Redundant vectors — those already in the span of the others — add nothing to the reach.
Spanning Sets
A set of vectors spans a vector space V if Span{v1,…,vk}=V — every vector in V can be written as a linear combination of the set. When this holds, the set is called a spanning set for V.
In Rn, a spanning set must contain at least n vectors. Fewer than n vectors cannot reach every direction — their span is a proper subspace of dimension less than n. A spanning set with exactly n independent vectors is a basis. A spanning set with more than n vectors contains redundancies that can be removed.
The standard basis {e1,…,en} spans Rn with no redundancy. The set {e1,e2,e1+e2} also spans R2, but the third vector is redundant — it lies in the span of the first two. Removing it leaves a basis.
Testing Whether a Vector Is in a Span
The question "Is b in Span{v1,…,vk}?" asks whether there exist scalars c1,…,ck such that c1v1+⋯+ckvk=b. This is a linear system: arrange the vectors as columns of a matrixA=[v1⋯vk] and check whether Ac=b is consistent.
Row reduce the augmented matrix [A∣b]. If a row of the form [00⋯0∣d] with d=0 appears, the system is inconsistent and b is not in the span.
Example: In the Span
Is b=(5,3,7) in Span{(1,0,1),(0,1,2)}? Form and reduce:
No contradiction. The solution is c1=5, c2=3, so (5,3,11)=5(1,0,1)+3(0,1,2).
Testing Whether a Set Spans Rⁿ
The question "Does {v1,…,vk} span Rn?" asks whether Ac=b is consistent for every possible right-hand side b∈Rn, where A is the matrix with the vectors as columns.
Row reduce A. The columns span Rn if and only if every row of the echelon form contains a pivot. If any row is entirely zero, then for some choices of b the augmented system will produce a [0⋯0∣d] contradiction, meaning those b are unreachable.
For the square case — n vectors in Rn — spanning reduces to a determinant test: the columns span Rn if and only if det(A)=0. This is equivalent to independence when the count matches the dimension, which is why the spanning and independence conditions coincide for sets of exactly n vectors in Rn.
Fewer than n vectors can never span Rn. An n×k matrix with k<n has at most k pivots, so at least n−k rows of the echelon form will be zero. More than n vectors can span Rn, but only if the set contains at least n independent vectors among them — the extras are redundant.
Span and Column Space
The column space of a matrix A is exactly the span of its columns:
Col(A)=Span{a1,a2,…,an}
This identity connects the abstract concept of span to the concrete question of system solvability. The system Ax=b has a solution if and only if b lies in the column space of A — that is, if and only if b is in the span of the columns.
The rank of A equals the dimension of the column space, which equals the number of independent columns. The pivot columns of the echelon form identify which original columns form a basis for the column space. The non-pivot columns are redundant — they lie in the span of the pivot columns.
This also explains why row reduction is the universal computational tool for span questions. Every question about span — membership, spanning, independence — translates into a question about the column space of some matrix, and row reduction answers all of them.
Redundancy and Reduction
A spanning set may contain more vectors than necessary. A vector vj in a spanning set is redundant if removing it does not shrink the span:
Span{v1,…,vk}=Span{v1,…,vj−1,vj+1,…,vk}
This happens exactly when vj is already a linear combination of the other vectors — it contributes no new directions.
Removing all redundant vectors from a spanning set produces a basis: a minimal spanning set with no waste. The process can be carried out systematically via row reduction. Arrange the vectors as columns, reduce, and identify the pivot columns. The original vectors corresponding to pivot positions form a basis for the span. The non-pivot columns are the redundant ones.
For example, if four vectors in R3 are given and row reduction reveals pivots in columns 1, 2, and 4, then the original first, second, and fourth vectors form a basis for the span. The third vector is a combination of the first two.
This reduction process is always possible in finite-dimensional spaces: every spanning set can be trimmed to a basis, and every independent set can be extended to one. The basis sits at the exact boundary between "too few vectors to span" and "too many vectors to be independent."
Span in Abstract Vector Spaces
The definition of span — the set of all linear combinations — applies in any vector space, not just Rn.
In the polynomial space P2, the span of {1,x,x2} is all of P2, since every polynomial a+bx+cx2 of degree at most 2 is a linear combination of these three. The set is independent and has the right count (3=dim(P2)), so it is the standard basis.
A subtler example is Span{1,sin2x,cos2x} in the space of continuous functions. This set has three elements, but the identity sin2x+cos2x=1 provides a dependence relation: 1⋅1+(−1)sin2x+(−1)cos2x=0. The span is therefore two-dimensional, not three-dimensional. Any two of the three functions form a basis for the span — the third is redundant.
In abstract spaces, the column-matrix approach is unavailable. Testing whether a specific vector lies in a span, or whether a set spans the whole space, requires working directly from the definition: write the target as a combination and check whether the resulting equation has a solution. The algebraic structure of the particular space — polynomial coefficients, function identities, matrix entries — determines how this check proceeds.