Question: Circle whether the following assertions are True or False:

1. T F A real, square matrix always has at least one real eigenvalue.
2. T F A finite dimensional vector space with an inner product always has an orthonormal basis.
3. T F Every real, symmetric matrix is diagonalizable.
4. \(\mathrm{T} \mathrm{F}\) If \(\mathrm{P}\) is an orthogonal matrix, then \(|\operatorname{det} P|=1\).
5. \(\mathrm{T} \mathrm{F}\) If \(\mathrm{u}\) is orthogonal to vectors \(\mathrm{v}\) and \(\mathrm{w}\), then \(\mathrm{u}\) is orthogonal to every linear combination of \(\mathrm{v}\) and \(\mathrm{w}\).

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