[Math] Prove an orthogonal map has orthogonal matrix with respect to orthonormal basis

linear algebra

Let $V$ be a vector space with the orthonormal basis $Q = \{ \vec{q_1},\ldots, \vec{q_n} \}$ and let $\ell:V\to V$ be an orthogonal map. Prove that the matrix $L$ of of $\ell$ with respect to $Q$ is orthogonal.

Note: By orthogonal map I mean that $\ell$ is linear and satisfies $\left\Vert \ell(\vec{x}) \right\Vert = \left\Vert \vec{x} \right\Vert$ for all $x \in V$. By orthogonal matrix I mean that $L$ has orthonormal columns.

I have that
$$
L=\left[
\begin{array}{ccc}
[\ell(\vec{q_1})]_Q & \cdots & [\ell(\vec{q_n})]_Q
\end{array}
\right]
$$

but I don't have any idea how to proceed.

Best Answer

A proof sketch.

  1. Show that $\langle \ell(\vec x), \ell(\vec {y}) \rangle = \langle \vec x , \vec y \rangle$ holds for all $\vec x, \vec y \in V$. For this step, you may need the following hint: $$\langle \vec x, \vec y \rangle = \frac{1}{2} (\| \vec x+\vec y \|^2 - \| \vec x \|^2 - \| \vec y \|^2) .$$

  2. Show that if the $i^{\rm th}$ column of $L$ is $c_i \in \mathbb R^n$, then $$ \ell(\vec {q_i}) = \sum_{k = 1}^n c_{i,k} \cdot \vec {q_k}. $$

  3. Show that for $1 \leq i, j \leq n$, we have $$ \langle \ell(\vec {q_i}) , \ell(\vec {q_j}) \rangle = \langle c_i, c_j \rangle. $$

  4. Using (1.), what can you say about $\langle \ell(\vec {q_i}) , \ell(\vec {q_j}) \rangle$? What does this mean about $\langle c_i, c_j \rangle$?

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