The linear span of a set of vectors is the collection of all vectors obtainable by taking their finite linear combinations. Equivalently, it is the smallest linear subspace containing the given set. A fundamental construction in linear algebra, span describes what a collection of vectors can generate using addition and scalar multiplication, whether those vectors are coordinate tuples, functions, or matrices. Common notation includes (\operatorname{span}(S)) and (\operatorname{span}{v_1,\ldots,v_k}). (math.brown.edu)
Definition and scalar field
Let (V) be a vector space over a field (F), and let (S\subseteq V). Its span is
[ \operatorname{span}F(S)= \left{ \sum{j=1}^{k}a_js_j: k\geq 0,\ a_j\in F,\ s_j\in S \right}. ]
The sum with (k=0) is interpreted as the zero vector. Consequently, (\operatorname{span}(\varnothing)={0}). For an infinite generating set, every individual vector in its span still uses only finitely many generators. Infinite series require additional structure and are not part of this algebraic definition. (cfm.brown.edu)
The scalar field matters. For example, regarding the complex numbers as a vector space over themselves, (\operatorname{span}{\mathbb C}{1}=\mathbb C). Over the real numbers, however, (\operatorname{span}{\mathbb R}{1}=\mathbb R), while ({1,i}) spans all of (\mathbb C). These follow directly from the definition by specifying the allowed coefficients. (cfm.brown.edu)
Smallest-subspace characterization
The span can also be defined through intersection:
[ \operatorname{span}(S)= \bigcap_{\substack{W\leq V\S\subseteq W}}W, ]
where (W\leq V) means that (W) is a linear subspace. The collection being intersected is nonempty because (V) itself contains (S). (cfm.brown.edu)
To see why the definitions agree, finite combinations contain zero and remain finite combinations after addition or scalar multiplication. They therefore form a subspace containing (S). Conversely, every subspace containing (S) must contain all such combinations by its closure properties. (cfm.brown.edu)
Several consequences follow: (S\subseteq\operatorname{span}(S)); if (S\subseteq T), then (\operatorname{span}(S)\subseteq\operatorname{span}(T)); and spanning an already generated subspace changes nothing. Adding a vector already in the span does not enlarge it. For subspaces (U,W), the span of their union is their sum (U+W={u+w:u\in U,w\in W}), although the union itself need not be a subspace. (cfm.brown.edu)
Geometric and functional examples
In real coordinate spaces, span has a direct geometric interpretation. A nonzero vector generates a line through the origin. Two vectors that are not scalar multiples generate a plane through the origin. Three linearly independent vectors in (\mathbb R^3) generate the whole space. Dependent generators may describe the same line or plane without adding new directions. (math.mit.edu)
For example,
[ \operatorname{span}{(1,0,0),(0,1,0)} ={(a,b,0):a,b\in\mathbb R}. ]
This is the coordinate plane (z=0). Adding ((1,1,0)) leaves the span unchanged because it is the sum of the original generators. (cfm.brown.edu)
Vectors need not be arrows. The polynomials (1,x,\ldots,x^n) span the space of polynomials of degree at most (n). The infinite set ({1,x,x^2,\ldots}) spans the space of all polynomials, not arbitrary infinite power series: each polynomial has only finitely many nonzero coefficients. (cfm.brown.edu)
Spanning sets, bases, and dimension
A set (S) is a spanning set for (V) when (\operatorname{span}(S)=V). Spanning and linear independence are distinct conditions: spanning concerns whether every target vector can be represented; independence concerns whether the generators have nontrivial linear relations. A basis satisfies both conditions and gives each vector a unique finite representation. (math.mit.edu)
A finite spanning set can be reduced to a basis by removing redundant vectors. Its span has dimension at most the number of generators, with equality precisely when they are linearly independent. Thus, an (n)-dimensional space requires at least (n) spanning vectors, and any spanning collection of exactly (n) vectors is a basis. Different bases can generate the same subspace. (math.mit.edu)
Matrices and computation
Place vectors (v_1,\ldots,v_k\in F^m) into the columns of a matrix (A). Then
[ Ax=x_1v_1+\cdots+x_kv_k, \qquad \operatorname{span}{v_1,\ldots,v_k} ={Ax:x\in F^k}. ]
Their span is the column space of (A), or the image of the linear map (x\mapsto Ax). Membership of a vector (b) in this span is equivalent to consistency of the system of linear equations (Ax=b). (math.mit.edu)
Gaussian elimination tests consistency and identifies pivot columns. The corresponding columns of the original matrix form a basis for its column space; their number is the rank. Row operations generally change the column space, so the basis vectors must be taken from the original matrix rather than its row-reduced version. (math.brown.edu)
For real matrices, least squares finds the closest attainable vector (Ax) to a target (b). This fitted vector is the orthogonal projection of (b) onto the column span. Even when dependent columns make the coefficient vector nonunique, the projected vector is unique. (live.ocw.mit.edu)
Related hull constructions
Linear span allows unrestricted scalar coefficients. The affine hull, associated with affine spaces, instead requires their sum to equal one. The convex hull, which generates a convex set, additionally requires nonnegative real coefficients. Accordingly, in a real vector space,
[ \operatorname{conv}(S)\subseteq \operatorname{aff}(S)\subseteq \operatorname{span}(S). ]
Unlike an affine hull, a linear span necessarily contains the origin. These constructions can therefore produce different sets from the same generators. (see.stanford.edu)