TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-16

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Every finite orthogonal list of nonzero vectors is linearly independent

Statement

Every finite orthogonal list of nonzero vectors in an inner product space is linearly independent. The empty list is included.

Facts & Assumptions

Given: An orthogonal list $(v_{0}, \dots, v_{r - 1})$ in an inner product space, with $v_{j} \neq 0$ for every $j < r$ .

[L1]

Orthogonality means $⟨ v_{i}, v_{j} ⟩ = 0$ whenever $i \neq j$ (Orthogonal vectors and subspaces, orthogonal and orthonormal sets, and orthonormal bases).

[L2]

Positive definiteness gives $⟨ v_{j}, v_{j} ⟩ > 0$ for every nonzero $v_{j}$ (Real and complex inner product spaces, with the inner product linear in the first argument).

[L3]

A finite list is linearly independent when its only vanishing linear combination has every coefficient zero (Linear independence: a finite list $v : n \to V$ is independent when $\sum_{i < n} λ_{i} v_{i} = 0_{V}$ forces every $λ_{i} = 0_{F}$ , and a subset $S \subseteq V$ is independent when every injective finite list into $S$ is independent).

Proof

technique · direct

1.1givenL3

If $r = 0$ , the independence condition is vacuous. Suppose $r > 0$ and $\sum_{i < r} a_{i} v_{i} = 0$ .

2.1step 1.1L1

For each $j < r$ , pair the equality in step 1.1 with $v_{j}$ . Linearity and [L1] give $0 = \sum_{i < r} a_{i} ⟨ v_{i}, v_{j} ⟩ = a_{j} ⟨ v_{j}, v_{j} ⟩$ .

3.1step 2.1L2L3∎

By [L2], $⟨ v_{j}, v_{j} ⟩ \neq 0$ , so $a_{j} = 0$ . This holds for every $j$ , and [L3] proves independence.

Depends on

Used by

Dependency tree · two levels

15 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Sheldon Axler, Linear Algebra Done Right, 4th ed., result 6.25 (standard reference, not scraped)
Sergei Treil, Linear Algebra Done Wrong, Theorem 5.2.6 (standard reference, not scraped)