Basis And Dimension

Introduction

Basis and dimension are central to understanding vector spaces in linear algebra. They provide minimal generating sets and quantify the size of spaces. Basis vectors uniquely represent every element of the space. Dimension is the cardinality of any basis, invariant under isomorphism.

"In mathematics, the notion of a basis captures the essence of linear structure and dimension measures its complexity." -- Gilbert Strang

Vector Spaces Overview

Definition

A vector space V over a field F is a set with two operations: vector addition and scalar multiplication satisfying closure, associativity, commutativity, identity, inverse, distributivity, and scalar identity axioms.

Examples

Examples include ℝⁿ, polynomial spaces, function spaces, and solution sets of homogeneous linear systems.

Subspaces

Subspace W ⊆ V is a subset closed under vector addition and scalar multiplication, containing the zero vector.

Linear Independence

Definition

A set of vectors {v₁, v₂, ..., vₖ} is linearly independent if no nontrivial linear combination equals zero: ∑ aᵢvᵢ = 0 implies all aᵢ = 0.

Testing Independence

Method: form matrix with vectors as columns, reduce to row echelon form; independence indicated by pivot in each column.

Significance

Independence ensures uniqueness of representation in terms of basis vectors.

Spanning Sets

Definition

A set S spans V if every vector in V is a linear combination of vectors in S.

Minimal Spanning Sets

Removing any vector from a minimal spanning set causes it to no longer span V; such sets are bases.

Examples

Standard unit vectors in ℝⁿ span ℝⁿ; polynomials of degree ≤ n span polynomial space Pₙ.

Definition of Basis

Formal Definition

A basis of vector space V is a linearly independent set that spans V.

Uniqueness of Representation

Every vector v ∈ V can be uniquely expressed as a linear combination of basis vectors.

Examples

Standard basis in ℝ³: {(1,0,0), (0,1,0), (0,0,1)}; polynomial basis: {1, x, x², ..., xⁿ} for Pₙ.

Definition of Dimension

Dimension as Cardinality

The dimension of V, dim(V), is the number of vectors in any basis of V.

Finite and Infinite Dimensions

Finite-dimensional spaces have finite bases; infinite-dimensional spaces do not.

Invariance

All bases of a vector space have the same cardinality; dimension is well-defined.

Properties of a Basis

Exchange Lemma

Given two bases, one vector in one can be exchanged with another, preserving basis status.

Extension Theorem

Any linearly independent set can be extended to a basis.

Reduction Theorem

Any spanning set can be reduced to a basis by removing dependent vectors.

Dimension Theorem

Statement

If U is a subspace of V and both are finite-dimensional, then dim(U) ≤ dim(V).

Dimension Formula

For subspaces U and W of V: dim(U + W) = dim(U) + dim(W) - dim(U ∩ W).

Rank-Nullity Theorem

For linear transformation T: V → W, dim(V) = rank(T) + nullity(T).

dim(V) = rank(T) + nullity(T)

Coordinate Systems and Bases

Coordinates Relative to Basis

Each vector v ∈ V corresponds to a unique n-tuple of scalars (its coordinates) relative to a chosen basis.

Change of Basis

Coordinate vectors transform via invertible matrices when changing from one basis to another.

Isomorphism with Fⁿ

Finite-dimensional vector spaces are isomorphic to Fⁿ via choice of basis.

Subspaces and Their Bases

Subspace Basis

Every subspace has a basis; dimension is the size of that basis.

Intersection and Sum

Subspaces intersect and sum to form new subspaces with dimensions governed by the dimension formula.

Examples

Kernel and image of linear maps are subspaces with bases and dimensions.

Computing a Basis

Method 1: Row Reduction

Form matrix with vectors as columns, apply Gaussian elimination, select pivot columns as basis vectors.

Method 2: Gram-Schmidt Process

Orthogonalizes vectors to produce an orthonormal basis from any linearly independent set.

Algorithmic Steps

1. Arrange vectors as columns in matrix A.2. Apply row operations to reach echelon form.3. Identify pivot columns; corresponding original vectors form basis. 

Applications of Basis and Dimension

Coordinate Representation

Basis allows vector representation as coordinate tuples, enabling computations in ℝⁿ.

Dimension in System Solutions

Dimension of solution space indicates degrees of freedom in linear systems.

Functional Analysis

Basis concepts extend to infinite dimensions in Hilbert and Banach spaces.

References

• Axler, S., Linear Algebra Done Right, Springer, 3rd ed., 2015, pp. 45-90.
• Strang, G., Introduction to Linear Algebra, Wellesley-Cambridge Press, 5th ed., 2016, pp. 100-145.
• Halmos, P. R., Finite-Dimensional Vector Spaces, Springer, 2nd ed., 1974, pp. 20-60.
• Friedberg, S. H., Insel, A. J., Spence, L. E., Linear Algebra, Prentice Hall, 4th ed., 2003, pp. 70-120.
• Lang, S., Linear Algebra, Springer, 3rd ed., 1987, pp. 30-80.