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Suppose e₁, ..., em is an orthonormal list of vectors in V. Let v € V. Prove that ||v||² = |(v, e₁) |² + ... + |(v, em) |² if and only if v € span(e₁, ..., em).

Suppose e₁, ..., em is an orthonormal list of vectors in V. Let v € V. Prove that ||v||² = |(v, e₁) |² + ... + |(v, em) |² if and only if v € span(e₁, ..., em).

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter4: Vector Spaces
Section4.2: Vector Spaces
Problem 43E: Prove that in a given vector space V, the zero vector is unique.
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It is from Linear Algebra Done Right by Axler.

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Suppose e₁, ..., em is an orthonormal list of vectors in V. Let v € V. Prove that ||v||² = |(v, e₁)|² + ... + |(v₂ em) | ² if and only if ve span(e₁, ..., em).
Transcribed Image Text:Suppose e₁, ..., em is an orthonormal list of vectors in V. Let v € V. Prove that ||v||² = |(v, e₁)|² + ... + |(v₂ em) | ² if and only if ve span(e₁, ..., em).
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