TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck 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

1.1

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

givenL3

2.1

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

step 1.1L1

3.1

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

step 2.1L2L3∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 49 results over 17 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click 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)