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Diagonalization Problems and Solutions in Linear Algebra


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  • Orthogonal Nonzero Vectors Are Linearly IndependentOrthogonal Nonzero Vectors Are Linearly Independent Let $S=\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k\}$ be a set of nonzero vectors in $\R^n$. Suppose that $S$ is an orthogonal set. (a) Show that $S$ is linearly independent. (b) If $k=n$, then prove that $S$ is a basis for $\R^n$.   Proof. (a) […]
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  • There is at Least One Real Eigenvalue of an Odd Real MatrixThere is at Least One Real Eigenvalue of an Odd Real Matrix Let $n$ be an odd integer and let $A$ be an $n\times n$ real matrix. Prove that the matrix $A$ has at least one real eigenvalue.   We give two proofs. Proof 1. Let $p(t)=\det(A-tI)$ be the characteristic polynomial of the matrix $A$. It is a degree $n$ […]
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