9. Let T be an operator on an n-dimensional vector space V over a field F. (a) If T has n distinct eigenvalues prove that T is diagonalizable. If T is diagonaziable prove that the characteristic polynomial of T splits over F. (c) Give an example of an operator T such that the characteristic polynomial of T splits but T is not diagonalizable.

Q: A*(x) = n(t, Un)un, x≤ H. Since for all x H, A* (A(x)) A(A*(x)) = Σkn(A(x), Un)Un = Σ|kn|²(x, Un)Un,…

A: Kindly refer to Step 2 for the solution.

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A: To work out the required properties of the linear operator f.

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A: The quadratic approximation of a function f at x0 is: f(x0)+f'(x0)(x-x0)+f''(x0)(x-x0)22 = Q(x)

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A: Using the Gram-Schmidt orthogonalisation prosess to find an orthogonal basis.

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A: 4) Solution:Given that, Now find the following:

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Q: W= {x: where x are eigenvectors of an A } Is W a subspace of R" ? Show the prove

A: See solution below

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A: The given mapping is given by:Tz=z¯We need to finda the matrix of T relative to the usual basis 1,ib…