Vector Spaces and Bases

Summary

A vector space is a set closed under addition and scalar multiplication with the usual algebraic axioms. Bases give coordinate systems: every vector has unique coordinates, and dimension is the number of basis vectors.

Prerequisites

Matrices and Row Reduction, Systems of Linear Equations. Hub: Linear Algebra.

Definition / Statement

A vector space $V$ over $\mathbb{R}$ is a set with addition and scalar multiplication satisfying the standard axioms (associativity, commutativity of addition, zero vector, additive inverses, distributive laws, $1\cdot \mathbf{v}=\mathbf{v}$ ).

Common examples: ${\mathbb{R}}^n$ , matrix spaces ${\mathbb{R}}^{m\times n}$ , polynomial spaces, solution spaces of homogeneous linear systems (null spaces).

A set $\{{\mathbf{v}}_1,\dots ,{\mathbf{v}}_k\}$ is linearly independent if

Otherwise it is linearly dependent.

The span of $\{{\mathbf{v}}_1,\dots ,{\mathbf{v}}_k\}$ is the set of all linear combinations of those vectors.

A basis of $V$ is a linearly independent spanning set. All bases of a finite-dimensional space have the same number of vectors; that number is $\dim V$ .

Objects and Dimensions

Notation

Conditions / Assumptions

• Real scalars unless complex spaces are specified.
• Finite-dimensional discussion unless stated otherwise.
• Rank–nullity for $A\in {\mathbb{R}}^{m\times n}$ : $$\dim \mathrm{Nul}(A)+\mathrm{rank}(A)=n.$$

Matrix / Vector Form

If $\mathcal{B}=\{{\mathbf{b}}_1,\dots ,{\mathbf{b}}_n\}$ is a basis of ${\mathbb{R}}^n$ and $B$ is the matrix with those columns, then

Pivot columns of $A$ form a basis for $\mathrm{Col}(A)$ . Free-variable special solutions form a basis for $\mathrm{Nul}(A)$ .

Geometric Interpretation

• Independent vectors point in “essentially different” directions (no one is a combination of the others).
• A basis is a coordinate frame for the whole space.
• Null space: directions mapped to $\mathbf{0}$ by $A$ ; column space: reachable right-hand sides $\mathbf{b}$ .

Worked Example

In ${\mathbb{R}}^3$ , let ${\mathbf{v}}_1=⟨1,0,0⟩$ , ${\mathbf{v}}_2=⟨1,1,0⟩$ , ${\mathbf{v}}_3=⟨1,1,1⟩$ .

The matrix with these columns is upper triangular with nonzero diagonals after mild reduction, so they are independent and form a basis of ${\mathbb{R}}^3$ . Coordinates of ${\mathbf{e}}_3=⟨0,0,1⟩$ :

so $[{\mathbf{e}}_3{]}_{\mathcal{B}}=⟨0,-1,1⟩$ .

Common Mistakes

• Thinking any set of $n$ vectors in ${\mathbb{R}}^n$ is a basis (must also be independent).
• Confusing $\mathrm{Col}(A)$ (in ${\mathbb{R}}^m$ ) with $\mathrm{Row}(A)$ (in ${\mathbb{R}}^n$ ).
• Claiming dependent sets cannot span a space (they can; bases are independent spanning sets).

Connections

• Related: Matrices and Row Reduction, Orthogonality and Projections, Eigenvalues and Eigenvectors
• Next: Determinants or Eigenvalues and Eigenvectors
• Geometry: directions and planes in Lines and Planes

References

Vector spaces, bases, and rank–nullity follow MIT 18.06.^[1]