If \(\mathbf{b}=(v_1,\ldots,v_n)\) is an orthonormal basis of a Euclidean space \((V,\langle\cdot{,}\cdot\rangle)\) and \(f:V\to V\) is an endomorphism, then \(\mathbf{c}=(f(v_1),\ldots,f(v_n))\) is an orthogonal basis of \((V,\langle\cdot{,}\cdot\rangle)\) if and only if \(f\) is an orthogonal transformation.

• True
• False

If \((V,\langle\cdot{,}\cdot\rangle)\) is a finite-dimensional Euclidean space and \(f:V\to V\) is an orthogonal transformation, then \(\mathbf{M}(f,\mathbf{b},\mathbf{c})\) is an orthogonal matrix for every choice of ordered bases \(\mathbf{b}\) and \(\mathbf{c}\) of \(V.\)

• True
• False

If \((V,\langle\cdot{,}\cdot\rangle)\) is a finite-dimensional Euclidean space and \(f:V\to V\) is an orthogonal transformation, then \(\|v-w\| = \|f(v)-f(w)\|\) for all \(v,w\in V.\)

• True
• False

If \((V,\langle\cdot{,}\cdot\rangle)\) is a finite-dimensional Euclidean space and \(f:V\to V\) is such that \(\|v-w\| = \|f(v)-f(w)\|\) for all \(v,w\in V.\) Then \(f\) must be an orthogonal transformation.

• True
• False

If \((V,\langle\cdot{,}\cdot\rangle)\) is a finite-dimensional Euclidean space and \(f:V\to V\) is an orthogonal transformation, then \(f\) must have a real eigenvalue.

• True
• False

If \((V,\langle\cdot{,}\cdot\rangle)\) is a finite-dimensional Euclidean space and \(f:V\to V\) is an orthogonal transformation, then \(f\) is self-adjoint.

• True
• False

If \((V,\langle\cdot{,}\cdot\rangle)\) is a finite-dimensional Euclidean space and \(f:V\to V\) is an orthogonal transformation, which is self-adjoint, then \(f=\mathrm{Id}_V.\)

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• False

If \((V,\langle\cdot{,}\cdot\rangle)\) is a finite-dimensional Euclidean space and \(f:V\to V\) is an orthogonal transformation, which is self-adjoint, then \(f^{-1}=f.\)

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• False

There exist matrices \(\mathbf{R},\mathbf S\in \mathrm{O}(n)\) such that \(\mathbf{R}\mathbf S\in\mathrm{SO}(n).\)

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• False

The only symmetric matrix in \(M_{2,2}(\mathbb{R})\) which is also in \(\mathrm{SO}(2)\) is given by \(\mathbf{1}_{2}.\)

• True
• False