Vector Spaces, Span, and Basis
Linear algebra operates within vector spaces — collections of objects that can be added together and scaled by numbers. Understanding what qualifies as a vector space, and what span and basis mean inside one, is the conceptual foundation for every other topic in linear algebra.
Vector Spaces: The Eight Axioms
A vector space over the real numbers is a set V with two operations — addition of vectors and multiplication of a vector by a real scalar — satisfying:
- Associativity of addition: u + (v + w) = (u + v) + w
- Commutativity of addition: u + v = v + u
- Zero vector: there exists 0 ∈ V such that v + 0 = v for all v
- Additive inverses: for every v there exists −v such that v + (−v) = 0
- Scalar associativity: a(bv) = (ab)v
- Identity scalar: 1·v = v
- Distributivity over vector addition: a(u + v) = au + av
- Distributivity over scalar addition: (a + b)v = av + bv
The most concrete example is ℝⁿ: ordered lists of n real numbers, with component-wise addition and scaling. Arrows in 2D or 3D space are another valid model, and the two are interchangeable via coordinates.
Linear Combinations and Span
A linear combination of vectors v₁, …, vₖ is any expression of the form:
The span of {v₁, …, vₖ} is the set of all such linear combinations. Geometrically:
- span{v}: a line through the origin in the direction of v (or just the origin if v = 0).
- span{u, v}: a plane through the origin when u and v point in different directions; a line if they are parallel.
- span{u, v, w} in ℝ³: all of ℝ³ if no vector lies in the span of the other two.
In ℝ², the standard basis vectors e₁ = (1,0) and e₂ = (0,1) satisfy span{e₁, e₂} = ℝ²: every vector (x, y) = x·e₁ + y·e₂.
Linear Independence
Vectors v₁, …, vₖ are linearly independent if the only solution to:
is a₁ = a₂ = ··· = aₖ = 0. Otherwise they are linearly dependent.
Intuitively, a set is linearly dependent when one of its vectors is redundant — it can be expressed as a linear combination of the others. Removing it does not shrink the span.
Basis and Dimension
A basis of V is a set of vectors that is:
- Linearly independent — no vector is redundant.
- Spanning — the span of the set is all of V.
Every basis of a finite-dimensional vector space contains the same number of vectors, called the dimension of V. For ℝⁿ, the dimension is n.
Standard bases: {(1,0), (0,1)} for ℝ², and {(1,0,0), (0,1,0), (0,0,1)} for ℝ³. Any two non-parallel 2D vectors form a valid (non-standard) basis for ℝ².
Worked Example: Checking Linear Independence
Are u = (2, 1), v = (1, 3) linearly independent in ℝ²?
Solve a·(2,1) + b·(1,3) = (0,0):
From the first equation b = −2a. Substituting: a + 3(−2a) = a − 6a = −5a = 0, so a = 0 and b = 0. The only solution is the trivial one.
Therefore {u, v} is linearly independent and forms a basis for ℝ². The determinant det([[2,1],[1,3]]) = 6 − 1 = 5 ≠ 0 confirms this.
Frequently Asked Questions
What is a vector space?
A vector space is a set V with two operations — vector addition and scalar multiplication — satisfying eight axioms: associativity and commutativity of addition, existence of a zero vector and additive inverses, compatibility of scalar multiplication with field multiplication, the identity scalar, and distributivity over both operations.
What is the span of a set of vectors?
The span of vectors v₁, v₂, …, vₖ is the set of all their linear combinations: {a₁v₁ + a₂v₂ + ··· + aₖvₖ | a₁,…,aₖ ∈ ℝ}. Geometrically, span{v} is a line through the origin, span{u,v} (when u and v are not parallel) is a plane through the origin.
What does it mean for vectors to be linearly independent?
Vectors v₁,…,vₖ are linearly independent if the only solution to a₁v₁ + ··· + aₖvₖ = 0 is a₁ = ··· = aₖ = 0. Intuitively, none of the vectors lies in the span of the others — each adds a genuinely new direction.
What is a basis?
A basis of a vector space V is a set of linearly independent vectors that span all of V. The number of vectors in any basis is always the same and is called the dimension of V. For example, ℝ² has dimension 2 and {(1,0),(0,1)} is its standard basis.
Adapted and rewritten from Silas Maths source notes; formulas reviewed for CalxSolver publication.