Linear Independence

A rigorous guide to linear independence, dependence relations, pivots, and proof strategies.

Vector-space ideas connect sets, operations, spanning, independence, basis, and dimension. A vector space sits at the centre of a dependency map. Subspace conditions lead to span and independence, which meet at basis and determine dimension; kernel and range connect through a transformation.

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.

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.