Need help on how to prove this numerical linear algebra problem.
1.)Let $Q \in M_n(\mathbb{R})$ and suppose that $\langle Qx, Qy \rangle = \langle x, y \rangle$ for every $x$, $y \in \mathbb{R}^n$. Prove that $Q$ is orthogonal.
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Note that $\langle x,y\rangle = x^T y$. Let $e_1,...,e_n$ be the standard basis vectors of $\mathbb{R}^n$. Then we have $$(Q^TQ)_{ij} = e_i^TQ^T Q e_j = (Qe_i)^T (Qe_j)= \langle Qe_i, Qe_j \rangle = \langle e_i, e_j\rangle = \delta_{ij}$$ where $\delta_{ij}$ is the Kronecker delta. And so $Q^TQ_ = I$, and so $Q$ is orthogonal.
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