Basis
Definition
A basis is a linearly independent spanning set. Every vector has a unique representation in terms of a basis.
| Property | Statement |
|---|---|
| Standard basis | e₁=(1,0,…,0), …, eₙ=(0,…,0,1) |
| Coordinates | [v]_B = coefficients in basis B |
Examples
Example 1. Is {(1,0),(0,1),(1,1)} a basis for ℝ²?
Solution. No — three vectors in ℝ² are linearly dependent.
In Depth
A basis of a vector space \(V\) is a linearly independent set that spans \(V\). Every vector in \(V\) has a unique representation as a linear combination of basis vectors. The number of basis vectors is the dimension of \(V\), which is the same for all bases (the dimension theorem).
The standard basis of \(\mathbb{R}^n\) is \(\{\mathbf{e}_1,\ldots,\mathbf{e}_n\}\). Other useful bases: eigenvector bases (diagonalize linear maps), orthonormal bases (simplify inner products), and wavelet bases (localize in both time and frequency). The choice of basis affects computational efficiency.
Change of basis: if \(B=\{\mathbf{b}_1,\ldots,\mathbf{b}_n\}\) is a basis, the change-of-basis matrix \(P=[\mathbf{b}_1|\cdots|\mathbf{b}_n]\) converts coordinates: \([\mathbf{v}]_{\text{std}}=P[\mathbf{v}]_B\). A linear map \(T\) has matrix \(P^{-1}AP\) in basis \(B\) if it has matrix \(A\) in the standard basis.
Ordered bases and coordinates: fixing an ordered basis \(B\) gives an isomorphism \(V\cong\mathbb{R}^n\) via coordinate vectors. This is why abstract vector spaces can be studied concretely using matrices — every finite-dimensional vector space over \(\mathbb{R}\) is isomorphic to \(\mathbb{R}^n\) for some \(n\).
In numerical analysis, the choice of basis affects conditioning. Monomial bases for polynomials are ill-conditioned; Chebyshev bases are much better. In signal processing, the Fourier basis diagonalizes convolution operators, making frequency-domain analysis efficient via the FFT.
Key Properties & Applications
The Gram–Schmidt process converts any linearly independent set \(\{\mathbf{v}_1,\ldots,\mathbf{v}_k\}\) to an orthonormal set \(\{\mathbf{q}_1,\ldots,\mathbf{q}_k\}\) spanning the same subspace. Modified Gram–Schmidt is numerically more stable and is equivalent to computing the thin QR decomposition.
Dual bases: for a basis \(B=\{\mathbf{b}_1,\ldots,\mathbf{b}_n\}\), the dual basis \(B^*=\{\mathbf{b}_1^*,\ldots,\mathbf{b}_n^*\}\) satisfies \(\langle\mathbf{b}_i^*,\mathbf{b}_j\rangle=\delta_{ij}\). For orthonormal bases, the dual basis equals the original. Dual bases are used in tensor analysis and finite element methods.
Lattice bases in cryptography: a lattice is a discrete subgroup of \(\mathbb{R}^n\), equivalently the set of integer linear combinations of basis vectors. The shortest vector problem (SVP) and closest vector problem (CVP) are hard lattice problems that underpin post-quantum cryptographic schemes.
Further Reading & Context
The study of basis connects to many areas of mathematics and its applications. Understanding the foundational definitions and theorems provides the basis for advanced work in analysis, algebra, and applied mathematics.
Historical development: most mathematical concepts evolved over centuries, with contributions from mathematicians across many cultures. The modern axiomatic treatment provides rigor, while computational tools enable practical application.
In modern mathematics, this topic appears in graduate courses and research across pure and applied mathematics. Connections to computer science, physics, and engineering make it a versatile and important area of study. Mastery of the core results and techniques opens doors to research in number theory, analysis, geometry, and beyond.
Recommended next steps: work through the standard theorems with full proofs, explore the connections to related topics listed above, and practice with a variety of problems ranging from computational exercises to theoretical proofs. The interplay between different areas of mathematics is one of the subject's greatest rewards.
Deep Dive: Basis
This lesson extends core ideas for basis with rigorous reasoning, edge-case checks, and application framing in linear algebra.
Practice Set
Practice. Derive one main result on this page and validate with a numeric or geometric check.
Goal. Confirm assumptions, transformation steps, and final interpretation.
References & Editorial Notes
- Stewart, Calculus.
- Strang, Introduction to Linear Algebra.
- Apostol, Mathematical Analysis.
Last editorial review: 2026-04-14.