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.
Key Formula
c1*v1 + c2*v2 + ... + cn*vn = 0 implies all ci = 0
LaTeX: c_1 v_1 + c_2 v_2 + \cdots + c_n v_n = \mathbf{0} \implies c_1 = c_2 = \cdots = c_n = 0
| Symbol | Meaning | Unit |
|---|---|---|
| v_i | i-th vector in the set | — |
| c_i | Scalar coefficient | — |
| 0 | Zero vector | — |
Worked Example
Problem
Determine whether the vectors v₁ = (1, 2, 3), v₂ = (0, 1, 4), v₃ = (2, 5, 10) are linearly independent.
Solution
Step 1: Set up c₁(1,2,3) + c₂(0,1,4) + c₃(2,5,10) = (0,0,0). Step 2: Write the system: c₁ + 2c₃ = 0 2c₁ + c₂ + 5c₃ = 0 3c₁ + 4c₂ + 10c₃ = 0 Step 3: From equation 1: c₁ = −2c₃. Step 4: Substitute into equation 2: −4c₃ + c₂ + 5c₃ = 0 → c₂ = −c₃. Step 5: Check equation 3: 3(−2c₃) + 4(−c₃) + 10c₃ = −6c₃ − 4c₃ + 10c₃ = 0. ✓ Step 6: Since c₃ is free (say c₃ = 1), we get c₁ = −2, c₂ = −1, c₃ = 1 as a non-trivial solution.
Answer
The vectors are linearly dependent; v₃ = 2v₁ − v₂ (verify: 2(1,2,3) − (0,1,4) = (2,3,2) ≠ v₃). Actually v₃ = 2v₁ + v₂ check: 2(1,2,3)+(0,1,4) = (2,5,10) ✓.
Linear Independence vs. Dependence: Key Comparisons
| Property | Linearly Independent | Linearly Dependent |
|---|---|---|
| Trivial solution only | Yes | No |
| Can form a basis | Yes (if spanning) | No |
| Determinant of matrix | Non-zero | Zero |
| Rank | Equals number of vectors | Less than number of vectors |
| Geometric interpretation | Vectors not coplanar/collinear | Vectors coplanar/collinear |
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.
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.
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 "linearis" (of a line) and "independere" (not to hang from). The formal concept was developed as part of Grassmann's "Ausdehnungslehre" (1844) and later formalised by Peano and others in the late 19th century.