If $Q$ is an orthogonal matrix, then the inner product $\langle Q u, Q v \rangle$ of transformed vectors $Q u$ and $Q v$ equals:
Options:
(A) $\langle u, v \rangle$
(B) $\frac{1}{2} \langle u, v \rangle$
(C) $\|u\| \|v\| \det(Q)$
(D) $0$
Answer Key:
A
Step-by-Step Solution & Explanation
Orthogonal transformations preserve inner products and lengths: $\langle Q u, Q v \rangle = (Q u)^T (Q v) = u^T Q^T Q v = u^T I v = u^T v = \langle u, v \rangle$.