Prove that multiplying vectors by an orthogonal matrix does not change their inner product or their length.

Vault Of Euclid

Orthogonal Matrices Preserve Inner Products and Norms

Inner product spaces and orthogonality warmup

Suppose \( P \) is an orthogonal matrix. Show that (a) \( \langle Pu, Pv \rangle = \langle u, v \rangle \) for any \( u, v \in V \); (b) \( \|Pu\| = \|u\| \) for every \( u \in V \). Use \( P^T P = I \) and \( \langle u, v \rangle = u^T v \).

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What you need first

Matrix multiplication
Matrix transpose properties
Definition of an orthogonal matrix
Relationship between inner product and vector norm

What this problem tests

Tests the ability to manipulate matrix transposes and apply the definitions of orthogonal matrices and vector norms.

This is a foundational result in linear algebra, establishing that orthogonal transformations correspond to rigid rotations and reflections.

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