Linear algebra

Linear independence

Vectors are linearly independent when no nontrivial linear combination of them equals zero.

Ask the Linear algebra assistant 1 min read · Updated September 9, 2026

Definition

Vectors are linearly independent when no nontrivial linear combination of them equals zero.

Intuition

Each independent vector adds a direction that the others cannot supply.

Worked example

(1, 0) and (0, 1) are independent. Adding (1, 1) makes the set dependent because it is their sum.

The math

∑icivi=0\sum_i c_i v_i=0 implies every ci=0c_i=0 for an independent set.

In machine learning

Independence identifies redundant features and determines whether coordinates in a spanning set are unique.

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Sources

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