Let $Q$ be an $n \times n$ real orthogonal matrix. For any two real vectors $x, y \in \mathbb{R}^n$, the standard inner product $\langle Qx, Qy \rangle$ equals:
Options:
(A) $\langle x, y \rangle$
(B) $-\langle x, y \rangle$
(C) $\det(Q) \langle x, y \rangle$
(D) $0$
Answer Key:
A
Step-by-Step Solution & Explanation
By definition of the matrix inner product:
$\langle Qx, Qy \rangle = (Qx)^T (Qy) = x^T (Q^T Q) y$
Since $Q$ is orthogonal, $Q^T Q = I$:
$\langle Qx, Qy \rangle = x^T I y = x^T y = \langle x, y \rangle$
Thus orthogonal transformations preserve inner products, vector lengths, and angles.