# Basis (linear algebra)
A basis of a vector space is a set of vectors that is linearly independent and spans the space, so that every vector in the space can be written in exactly one way as a linear combination of the basis vectors[1](https://archive.nptel.ac.in/content/storage2/courses/122104018/node39.html). The basis is the device that turns abstract vectors into coordinate lists: once a basis is fixed, vector arithmetic becomes arithmetic on the coefficients[2](https://www.cs.yale.edu/homes/aspnes/pinewiki/attachments/LinearAlgebra/linear-algebra.pdf).
| Key fact | Detail |
|---|---|
| Definition | A subset \( \mathcal{B} \) of a vector space \( V \) is a basis if it is linearly independent and \( L(\mathcal{B}) = V \), that is, every vector of \( V \) is a linear combination of elements of \( \mathcal{B} \)[1](https://archive.nptel.ac.in/content/storage2/courses/122104018/node39.html) |
| Unique representation | If \( \{x_i\} \) is a basis of \( V \), every \( y \in V \) has a unique representation \( y = a_1 x_1 + a_2 x_2 + \cdots + a_n x_n \)[2](https://www.cs.yale.edu/homes/aspnes/pinewiki/attachments/LinearAlgebra/linear-algebra.pdf) |
| Well-defined size | If a vector space has a basis with \( n \) vectors, then every basis has \( n \) vectors[3](https://pressbooks.cuny.edu/mat5600linearalgebra/chapter/basis-and-dimension/) |
| Dimension | The dimension of \( V \) is the number of vectors in a basis, with \( \dim(\{\mathbf{0}\}) = 0 \); for example \( \dim(\mathbb{R}^n) = n \), \( \dim(M_{m \times n}) = mn \), and \( \dim(\mathcal{P}_n) = n+1 \)[4](https://ximera.osu.edu/linearalgebra/textbook/describingVectorSpaces/basisAndDimension) • [3](https://pressbooks.cuny.edu/mat5600linearalgebra/chapter/basis-and-dimension/) |
| Existence | Every vector space has a basis, a result whose general form relies on the axiom of choice; any two bases of an infinite-dimensional space have the same cardinality[5](https://courses.math.wichita.edu/math511/ch4/5/BasisAndDimension.html) |
| Standard basis | The vectors \( \mathbf{e}_i \) with 1 in the \( i \)-th place and 0 elsewhere form the standard basis of \( \mathbb{R}^n \)[1](https://archive.nptel.ac.in/content/storage2/courses/122104018/node39.html) |
| Infinite-dimensional variant | A Schauder basis allows convergent infinite series of basis vectors, while a Hamel basis allows only finite linear combinations[6](https://web.math.princeton.edu/~js129/PDFs/teaching/MAT520_fall_2025/MAT520_Lecture_Notes.pdf) |
## Definition and unique coordinates
The two defining conditions work together. [Linear independence](https://www.edgepedia.org/linear-independence) means no nontrivial finite linear combination of the basis vectors equals zero; spanning means the set generates the whole space. A set that spans but is not independent gives multiple representations of the same vector, and a set that is independent but does not span misses some vectors entirely[1](https://archive.nptel.ac.in/content/storage2/courses/122104018/node39.html).
Uniqueness is the property that makes bases useful. If a vector \( \mathbf{v} \) had two representations \( \sum \alpha_i \mathbf{v}_i = \sum \beta_i \mathbf{v}_i \), subtracting gives a linear combination of basis vectors equal to zero, so independence forces \( \alpha_i = \beta_i \) for every \( i \)[1](https://archive.nptel.ac.in/content/storage2/courses/122104018/node39.html). The resulting coefficients are called the coordinates of \( \mathbf{v} \) relative to the basis, written \( [\mathbf{v}]_B \)[7](https://pressbooks.cuny.edu/mat5600linearalgebra/chapter/coordinates-and-change-of-basis/). With coordinates fixed, vector addition and scalar multiplication reduce to operations on the coefficient lists[2](https://www.cs.yale.edu/homes/aspnes/pinewiki/attachments/LinearAlgebra/linear-algebra.pdf).
A basis can also be characterized as a minimal spanning set: if a spanning set contained a vector expressible through the others, removing it would not change the span, and the set could not be independent[4](https://ximera.osu.edu/linearalgebra/textbook/describingVectorSpaces/basisAndDimension).
## Examples
- The standard basis of \( \mathbb{R}^n \) is \( \{\mathbf{e}_1, \ldots, \mathbf{e}_n\} \); for \( n = 3 \) this is \( \{(1,0,0), (0,1,0), (0,0,1)\} \)[1](https://archive.nptel.ac.in/content/storage2/courses/122104018/node39.html).
- Bases are not unique. In the plane \( \{(x,y,0)\} \subset \mathbb{R}^3 \), the sets \( \{(1,0,0),(0,1,0)\} \), \( \{(1,0,0),(1,1,0)\} \), and \( \{(2,0,0),(1,3,0)\} \) are all bases[1](https://archive.nptel.ac.in/content/storage2/courses/122104018/node39.html).
- The same set of vectors can carry different bases depending on the scalar field: \( \mathbb{C} \) as a complex vector space has basis \( \{\mathbf{1}\} \), while as a real vector space it has basis \( \{\mathbf{1}, \mathbf{i}\} \)[1](https://archive.nptel.ac.in/content/storage2/courses/122104018/node39.html).
- The vector space of all polynomials with real coefficients has the infinite basis \( \{1, x, x^2, \ldots\} \), making it infinite dimensional[1](https://archive.nptel.ac.in/content/storage2/courses/122104018/node39.html).
- A redundant set is not a basis: in the subspace \( x + y - z = 0 \) of \( \mathbb{R}^3 \), the vector \( (3,2,5) \) equals both \( (1,1,2) + (2,1,3) \) and \( 4(1,1,2) - (1,2,3) \), so the set \( \{(1,1,2), (2,1,3), (1,2,3)\} \) cannot be a basis[1](https://archive.nptel.ac.in/content/storage2/courses/122104018/node39.html).
## Dimension
All bases of a finite-dimensional space have the same number of elements, so the dimension \( \dim(V) \), defined as the number of vectors in any basis, is a property of the space alone[3](https://pressbooks.cuny.edu/mat5600linearalgebra/chapter/basis-and-dimension/). In full generality, any two bases of a nonzero vector space have the same cardinality, and dimension is described as the single most important numerical invariant one can attach to a vector space[4](https://ximera.osu.edu/linearalgebra/textbook/describingVectorSpaces/basisAndDimension).
This invariance yields counting constraints. In a space of dimension \( n \), any subset with more than \( n \) vectors is linearly dependent, and any \( n \) linearly independent vectors automatically span and hence form a basis[3](https://pressbooks.cuny.edu/mat5600linearalgebra/chapter/basis-and-dimension/). Two constructive facts follow: any linearly independent set can be extended to a basis by adding vectors, and any spanning set can be reduced to a basis by removing dependent vectors[5](https://courses.math.wichita.edu/math511/ch4/5/BasisAndDimension.html).
## Existence and the axiom of choice
For finite-dimensional spaces, a basis can be built directly: pick a nonzero vector, and as long as the current set does not span, add a vector outside its span; independence is preserved at each step and the process ends because the space is finite dimensional[1](https://archive.nptel.ac.in/content/storage2/courses/122104018/node39.html). For arbitrary vector spaces, the statement that every vector space has a basis is proved with [Zorn's lemma](https://www.edgepedia.org/zorns-lemma), which is equivalent to the axiom of choice[4](https://ximera.osu.edu/linearalgebra/textbook/describingVectorSpaces/basisAndDimension) • [8](https://ncatlab.org/nlab/show/basis%2Bin%2Bfunctional%2Banalysis).
Such bases can be far from explicit.
## Hamel and Schauder bases in infinite dimensions
In functional analysis the word basis splits into two notions. A Hamel basis is the algebraic one: every vector is a finite linear combination of basis vectors[6](https://web.math.princeton.edu/~js129/PDFs/teaching/MAT520_fall_2025/MAT520_Lecture_Notes.pdf). A Schauder basis instead requires a topology, usually a norm, and writes each vector as a convergent infinite series \( v = \sum_{n=1}^{\infty} f_n b_n \) with unique coefficients[6](https://web.math.princeton.edu/~js129/PDFs/teaching/MAT520_fall_2025/MAT520_Lecture_Notes.pdf) • [10](https://www.sciencedirect.com/topics/mathematics/additive-group).
The distinction matters because infinite-dimensional Banach spaces admit no countable Hamel basis, while a separable [Hilbert space](https://www.edgepedia.org/hilbert-space) does have a countable orthonormal Schauder basis[6](https://web.math.princeton.edu/~js129/PDFs/teaching/MAT520_fall_2025/MAT520_Lecture_Notes.pdf). A long-standing question, whether every separable infinite-dimensional [Banach space](https://www.edgepedia.org/banach-space) has a Schauder basis, was answered negatively, with Per Enflo constructing a counterexample[6](https://web.math.princeton.edu/~js129/PDFs/teaching/MAT520_fall_2025/MAT520_Lecture_Notes.pdf).
## Change of basis
Fixing a basis \( \mathcal{B} = \{\mathbf{b}_1, \ldots, \mathbf{b}_n\} \) of \( \mathbb{R}^n \), the change-of-coordinates matrix \( P_{\mathcal{B}} \) has the basis vectors as its columns. It is invertible, and coordinates convert by \( [\mathbf{v}]_{\mathcal{B}} = P_{\mathcal{B}}^{-1} \mathbf{v} \), while \( \mathbf{v} = P_{\mathcal{B}} [\mathbf{v}]_{\mathcal{B}} \)[11](https://interactivetextbooks.tudelft.nl/linear-algebra/Chapter4/ChangeOfBasis.html). For a linear transformation \( T \) with standard matrix \( A \), the matrix of \( T \) with respect to \( \mathcal{B} \) is \( [T]_{\mathcal{B}} = P^{-1} A P \)[11](https://interactivetextbooks.tudelft.nl/linear-algebra/Chapter4/ChangeOfBasis.html). Choosing a basis adapted to a problem, for example one aligned with a plane so that projection becomes simple, is often the practical point of the whole construction[11](https://interactivetextbooks.tudelft.nl/linear-algebra/Chapter4/ChangeOfBasis.html).
## Historical origin
The modern concept traces to Hermann Grassmann, the German mathematician and linguist (1809–1877). From 1832 onward he developed a vector calculus in which an \( n \)-dimensional space is spanned by a system of linearly independent units, the basis vectors, through which any other vector is represented by linear combination[12](https://mathshistory.st-andrews.ac.uk/Strick/grassmann.pdf). His 1844 *Ausdehnungslehre* also treated subspaces, and he proved the dimensional formula now written as \( \dim(U + W) = \dim U + \dim W - \dim(U \cap W) \)[12](https://mathshistory.st-andrews.ac.uk/Strick/grassmann.pdf). He further obtained the formula for change of coordinates under change of basis and showed that every change of basis, equivalently every invertible linear transformation, is a product of elementary transformations[13](https://mathshistory.st-andrews.ac.uk//Biographies/Grassmann/). The broader unification of linear questions around determinants, driven by the move beyond three dimensions in mid-19th-century geometry, preceded the fully abstract theory of vector spaces that followed[14](https://www.sciencedirect.com/science/article/pii/S0315086085710245).
## References
1. [Bases, NPTEL course notes](https://archive.nptel.ac.in/content/storage2/courses/122104018/node39.html)
2. [Notes on linear algebra, James Aspnes, Yale University](https://www.cs.yale.edu/homes/aspnes/pinewiki/attachments/LinearAlgebra/linear-algebra.pdf)
3. [Basis and Dimension, Linear Algebra: Lecture Notes, CUNY Pressbooks](https://pressbooks.cuny.edu/mat5600linearalgebra/chapter/basis-and-dimension/)
4. [Bases and dimension, Ximera, Ohio State University](https://ximera.osu.edu/linearalgebra/textbook/describingVectorSpaces/basisAndDimension)
5. [Basis and Dimension, Wichita State University Math 511](https://courses.math.wichita.edu/math511/ch4/5/BasisAndDimension.html)
6. [Functional Analysis MAT520 Lecture Notes, Princeton University](https://web.math.princeton.edu/~js129/PDFs/teaching/MAT520_fall_2025/MAT520_Lecture_Notes.pdf)
7. [Coordinates and Change of Basis, Linear Algebra: Lecture Notes, CUNY Pressbooks](https://pressbooks.cuny.edu/mat5600linearalgebra/chapter/coordinates-and-change-of-basis/)
8. [Basis in functional analysis, nLab](https://ncatlab.org/nlab/show/basis%2Bin%2Bfunctional%2Banalysis)
9. [On infinite-dimensional Banach spaces and weak forms of the axiom of choice, Tachtsis](https://luisto.fi/courses/MATS220/documents/AC-Banach.pdf)
10. [Additive Group, ScienceDirect Topics](https://www.sciencedirect.com/topics/mathematics/additive-group)
11. [Change of basis, Linear Algebra, TU Delft interactive textbook](https://interactivetextbooks.tudelft.nl/linear-algebra/Chapter4/ChangeOfBasis.html)
12. [Grassmann, MacTutor History of Mathematics (Strick biographical poster)](https://mathshistory.st-andrews.ac.uk/Strick/grassmann.pdf)
13. [Hermann Grassmann (1809–1877), MacTutor History of Mathematics biography](https://mathshistory.st-andrews.ac.uk//Biographies/Grassmann/)
14. [A general outline of the genesis of vector space theory, Historia Mathematica, ScienceDirect](https://www.sciencedirect.com/science/article/pii/S0315086085710245)
---
*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Linear and multilinear algebra › Vector spaces and linear maps*
*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*
*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*
License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license