Linear Independence

Precise definition

Vectors $v_1,\ldots,v_k$ are linearly independent if $c_1v_1+\cdots+c_kv_k=0$ implies $c_1=\cdots=c_k=0$. They are dependent if a nontrivial coefficient list produces zero, equivalently at least one vector is a combination of the others.

Notation and mathematical language

Place coordinate vectors as columns of $A$. Independence means $A c=0$ has only the trivial solution, so every column is a pivot column. In a function or polynomial space, the same definition applies; compare coefficients or evaluate strategically.

Conceptual picture

Independence means each direction contributes information that cannot be reconstructed from the others. A dependence relation reveals redundancy. The zero vector makes any list containing it dependent because coefficient 1 on zero already gives a nontrivial relation.

Conditions and key results

The ambient field matters, and the vectors must belong to the same vector space. More than $n$ vectors in an $n$-dimensional space are dependent. Fewer than or equal to $n$ may still be dependent, so counting alone gives only a one-way conclusion.

A reliable strategy

1. Write the homogeneous combination equal to zero in the correct ambient coordinates.
2. Solve for coefficients by row reduction, coefficient comparison, or an appropriate theorem.
3. If only the zero coefficient vector occurs, conclude independence; otherwise display a nontrivial relation.
4. Check the conclusion against dimension, zero vectors, duplicates, and scalar multiples.

Fully worked example

Interpretation and application

Independence determines unique coordinates, identifiable parameters, nonredundant features, and fundamental solution sets. Near dependence in numerical data can cause instability even when exact algebra says independent.

Common mistakes

Verification and reasonableness

• Substitute a proposed dependence relation and verify the zero vector.
• Compare pivot count with number of vectors.
• For square coordinate matrices, check determinant nonzero exactly matches independence.

Practice

1. Are $(1,2)$ and $(2,4)$ independent?
2. Can four vectors in $\mathbb R^3$ be independent?
3. Why does a list containing zero fail?

Further deduction

A set of eigenvectors belonging to distinct eigenvalues is independent. If $\sum c_iv_i=0$, applying $A-\lambda_kI$ removes the $k$th mode and creates a shorter relation among the others; induction forces all coefficients to zero. This theorem supplies independence without row reduction and is central to diagonalization.

A Wronskian can test independence of differentiable functions in appropriate differential-equation settings, but a zero Wronskian at one point does not prove general functional dependence without extra hypotheses. The definition through a constant-coefficient linear combination is always authoritative; specialized tests must be used within their theorems.

Functions can be tested directly from the definition by evaluating a proposed identity at strategically chosen inputs. If $c_0+c_1x+c_2x^2=0$ for every real $x$, substituting three distinct values yields a system whose only solution is $c_0=c_1=c_2=0$; alternatively, polynomial coefficient uniqueness gives the result immediately. A few numerical evaluations prove independence only when they create a nonsingular system and the assumed identity is known to hold throughout the domain.

Related topics

• Basis
• Span
• Dimension