A rotation in the plane about an angle \(\alpha\ne k\pi, k\in\mathbb Z\) is always normal but never self-adjoint.

• True
• False

Let \((V,\langle\cdot{,}\cdot\rangle)\) be a finite dimensional Euclidean space and \(f:V\to V\) be a self-adjoint endomorphism. If \(v\) is an eigenvector of \(f,\) then any other eigenvector \(w\) of \(f\) must satisfy \(\langle v,w\rangle=0.\)

• True
• False

Let \((V,\langle\cdot{,}\cdot\rangle)\) be a finite dimensional Euclidean space and \(f:V\to V\) be a self-adjoint endomorphism. Then any two eigenvectors with respect to the same eigenvalue must be parallel.

• True
• False

Let \((V,\langle\cdot{,}\cdot\rangle)\) be a finite dimensional Euclidean space and \(f:V\to V\) be a self-adjoint endomorphism. If \(v\) is an eigenvector with respect to \(\lambda\) and \(w\) is an eigenvector with respect to \(\mu\ne\lambda,\) then \(\langle v,w\rangle = 0.\)

• True
• False

Let \((V,\langle\cdot{,}\cdot\rangle)\) be a finite dimensional Euclidean space and \(f:V\to V\) be a self-adjoint endomorphism. Then \(V\) can admit distinct orthonormal bases of eigenvectors of \(f.\)

• True
• False

Let \((V,\langle\cdot{,}\cdot\rangle)\) be a finite dimensional Euclidean space and \(f:V\to V\) be a self-adjoint endomorphism. Then \(V\) can be written as an orthogonal direct sum of eigenspaces of \(f.\)

• True
• False

Let \((V,\langle\cdot{,}\cdot\rangle)\) be a finite dimensional Euclidean space and \(f:V\to V\) be a linear map such that \(f\circ f=f.\) Then \(f\) must be self-adjoint.

• True
• False

Let \((V,\langle\cdot{,}\cdot\rangle)\) be a finite dimensional Euclidean space and \(f:V\to V\) be a linear map such that \(f\circ f=f.\) Then \(f\) must be normal.

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• False

Let \(q\) and \(\widetilde q\) be two quadratic forms defined by the symmetric bilinear forms \(\langle\cdot{,}\cdot\rangle\) and \(\langle\!\langle\cdot{,}\cdot\rangle\!\rangle.\) If \(q-\widetilde q\) is the zero map, then \(\langle\cdot{,}\cdot\rangle\) and \(\langle\!\langle\cdot{,}\cdot\rangle\!\rangle\) agree.

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• False

Let \(q:\mathbb{R}^2\to\mathbb{R}\) be a quadratic form and let \(\mathbf{b}=(v_1,v_2)\) be an orthonormal ordered basis of \(\mathbb{R}^2\) such that \(q(v) = \boldsymbol{\beta}(v)^T \mathbf{D}\boldsymbol{\beta}(v),\) where \(\mathbf{D}\) is diagonal. Then the map \(\boldsymbol{\beta}:\mathbb{R}^2\to\mathbb{R}^2\) is an orthogonal transformation with respect to the inner product which defines \(q.\)

• True
• False