Span and Linear Combinations

Subtopic: Vector Spaces

A linear combination mixes vectors using addition and scalar multiplication—the two fundamental operations in a vector space. The span of a set of vectors is everything you can build from those vectors using linear combinations. Span tells you the "reach" of your vectors: what space they can fill.

Introduction

Given a few vectors, what can you make from them? If you can add them, subtract them, scale them, and combine those operations—what collection of vectors can you reach?

With a single nonzero vector v in (R^2), you can reach everything on the line through v and the origin. With two non-parallel vectors, you can reach any point in (R^2). That's span: the set of all destinations reachable by combining your starting vectors.

Definition

Linear Combination

A linear combination of vectors (v_1),(v_2),…,(v_k) is any expression of the form:

(c_1)*(v_1)+(c_2)*(v_2)+⋯+(c_k)*(v_k)

where (c_1),(c_2),…,(c_k) are scalars (the coefficients).

Span

The span of vectors (v_1),(v_2),…,(v_k) is the set of all their linear combinations: span{(v_1),(v_2),…,(v_k)}={(c_1)*(v_1)+(c_2)*(v_2)+⋯+(c_k)*(v_k):(c_1),…,(c_k)∈F}

The span is always a subspace—it contains 0 (take all coefficients zero), and it's closed under addition and scalar multiplication.

Geometric Interpretation

In (R^3):

• The span of one nonzero vector is a line through the origin

• The span of two non-parallel vectors is a plane through the origin

• The span of three vectors that don't all lie in one plane is all of (R^3)

Adding more vectors can expand the span (if they point in new directions) or leave it unchanged (if they're already in the span).

Think of each vector as opening a new degree of freedom. But if a new vector is already reachable from the others, it adds no new freedom—the span doesn't grow.

Worked Example 1

Let (v_1)=(1,0,2) and (v_2)=(0,1,1). What is span{(v_1),(v_2)}?

The span consists of all vectors (c_1)(1,0,2)+(c_2)(0,1,1)=((c_1),(c_2),2*(c_1)+(c_2)).

This is the set {(x,y,z):z=2*x+y} — a plane through the origin in (R^3).

To check if a specific vector is in the span, like (3,2,8):

We need (c_1)=3, (c_2)=2, and check 2*(3)+2=8

Yes, (3,2,8)=3*(v_1)+2*(v_2) is in the span.

Worked Example 2

Is (1,2,3) in span{(1,1,0),(0,1,1)} ?

We need to find (c_1),(c_2) such that (c_1)(1,1,0)+(c_2)(0,1,1)=(1,2,3).

This gives the system:

(c_1)=1
(c_1)+(c_2)=2
(c_2)=3

From equation 1: (c_1)=1.
From equation 3: (c_2)=3.
Check equation 2: 1+3=4≠2.

The system has no solution. Therefore (1,2,3) is NOT in the span.

Checking if a Vector is in a Span

To determine if a vector b is in span{(v_1),…,(v_k)}:

1. Set up the equation (c_1)*(v_1)+(c_2)*(v_2)+⋯+(c_k)*(v_k)=b.

2. Write this as a linear system (or matrix equation).

3. If the system has a solution, b is in the span. If not, it is not.

In matrix form: b is in span{(v_1),…,(v_k)} if and only if the system A*x=b has a solution, where A=[[(v_1),(v_2),…,(v_k)]] .

Spanning Sets

A set of vectors {(v_1),…,(v_k)} spans a vector space V if span{(v_1),…,(v_k)}=V — every vector in V can be written as a linear combination of the (v_i).

Examples:

• {(1,0),(0,1)} spans (R^2) (the standard basis)

• {(1,1),(1,−1)} also spans (R^2) (different basis)

• {(1,0),(2,0)} does NOT span (R^2) — both vectors lie on the x-axis, so the span is just a line

Key Properties

• The span of any set of vectors is a subspace (the smallest subspace containing those vectors).

• span{v}={c*v:c∈F} — a line through the origin (or {0} if v=0).

• Adding a vector to a set either expands the span or leaves it unchanged — it never shrinks.

• If v∈span{(v_1),…,(v_k)}, then span{(v_1),…,(v_k),v}=span{(v_1),…,(v_k)}.

• span{0}={0} — the zero vector alone spans only the trivial subspace.

Connection to Matrices

The column space of a matrix A is exactly the span of its columns. If A=[[(a_1),(a_2),…,(a_n)]], then: col(A)=span{(a_1),(a_2),…,(a_n)}

This means: A*x=b has a solution if and only if b is in the span of A columns.

Applications

In computer graphics, colors are often represented as linear combinations of basis colors (RGB). The span of {Red, Green, Blue} is all displayable colors.

In signal processing, any signal in a given space can be written as a linear combination of basis signals (like sine waves in Fourier analysis).

In data science, the span of feature vectors determines what relationships a linear model can capture.

Summary

A linear combination is a sum of scaled vectors. The span of a set of vectors is all their possible linear combinations—the subspace they generate. To check if a vector is in a span, solve the corresponding linear system. A spanning set for a space V means every vector in V can be expressed as a linear combination of the set. The concept connects directly to column spaces of matrices.