Write down detailed proofs of every statement you make
1. Let A be a real n × n matrix with an eigenvalue λ having algebraic multiplicity n. Prove that for any t real one has
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2. Let A denote the matrix
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• Find an orthogonal matrix O such that OT AO is diagonal
• Compute the matrix eA.
3. Consider the vector space of polynomials with real coefficients and withinner product
Apply the Graham-Schmidt process to find an orthonormal basis, with respect to this inner product, for the subspace generated by .
4. Let A be a real n × n matrix. Define . Find necessary and sufficient conditions on A for this operation to be a inner product on R3.
5. Show that the system Ax = b has no solution and find the least square solution of the problem Ax ≈ b with
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