Basis (Linear Algebra)
A basis of a vector space is a set of linearly independent vectors that spans the entire space, meaning every vector in the space can be expressed as a unique linear combination of the basis vectors. The number of vectors in any basis of a finite-dimensional vector space is the same and equals the dimension of the space. Bases are fundamental in coordinate systems, data compression, Fourier analysis, and the numerical solution of linear systems.
Key Formula
v = c1*b1 + c2*b2 + ... + cn*bn
LaTeX: v = c_1 b_1 + c_2 b_2 + \cdots + c_n b_n
| Symbol | Meaning | Unit |
|---|---|---|
| v | Any vector in the space | — |
| b_i | i-th basis vector | — |
| c_i | Unique scalar coordinate | — |
| n | Dimension of the vector space | — |
Worked Example
Problem
Show that {(1, 0), (0, 1)} is a basis for ℝ² and express the vector (3, −5) in terms of this basis.
Solution
Step 1: Check linear independence. Assume a(1,0) + b(0,1) = (0,0). This gives a = 0 and b = 0, so the vectors are linearly independent. Step 2: Check spanning. Any (x, y) = x(1,0) + y(0,1), so the set spans ℝ². Step 3: Since the set is linearly independent and spans ℝ², it is a basis. Step 4: Express (3, −5) = 3·(1,0) + (−5)·(0,1).
Answer
(3, −5) = 3·e₁ + (−5)·e₂; this is the standard basis representation.
Common Bases in Different Vector Spaces
| Vector Space | Standard Basis | Dimension |
|---|---|---|
| ℝ² | {(1,0), (0,1)} | 2 |
| ℝ³ | {(1,0,0), (0,1,0), (0,0,1)} | 3 |
| Polynomials ≤ degree 2 | {1, x, x²} | 3 |
| 2×2 matrices | {E11, E12, E21, E22} | 4 |
| Fourier (periodic fns) | {sin(nx), cos(nx)} | ∞ |
Interactive Tools
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Related Terms
Vector Space
A vector space is a set of objects called vectors, together with two operations — vector addition and scalar multiplication — that satisfy eight axioms including closure, associativity, distributivity, and the existence of zero and inverse elements. Vector spaces provide the foundational framework for linear algebra, generalising the familiar geometry of arrows in 2D and 3D to abstract settings of any dimension. They are essential in physics, engineering, computer graphics, machine learning, and quantum mechanics, where states, signals, and transformations are all described as elements of appropriate vector spaces.
Linear Independence
A set of vectors is linearly independent if no vector in the set can be expressed as a linear combination of the others, equivalently if the only solution to the equation c₁v₁ + c₂v₂ + … + cₙvₙ = 0 is the trivial solution where all scalars cᵢ equal zero. If at least one non-zero scalar solution exists, the set is called linearly dependent. Linear independence is the key criterion for determining whether a set of vectors forms a basis, and it underlies the concepts of dimension, rank, and solutions to linear systems.
Matrix Rank
The rank of a matrix is the dimension of the vector space spanned by its columns (column rank), which always equals the dimension spanned by its rows (row rank), giving a single well-defined measure of the "information content" or degrees of freedom in the matrix. A matrix A of size m×n has rank at most min(m, n); if the rank equals min(m, n) the matrix is called full rank. The rank determines whether a linear system Ax = b has a unique solution (full rank square matrix), infinitely many solutions, or no solution, making it central to the theory of linear equations, least-squares fitting, and dimensionality reduction.
From Latin "basis" (foundation, pedestal), itself from Greek "basis" (step, base). In mathematics, the term was introduced in the context of coordinate systems in the early 20th century by mathematicians formalising linear algebra.