Linear transformations, Matrix diagonalization and Orthogonal projections
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R stands for the field of real numbers. C stands for the field of complex numbers.
1. Let T be a linear transformation from the set P2(R) of all polynomials of degree at most 2 into itself.
T: P2(R) --> P2(R), given by T(f) = f' - f'', fEP2(R),
where f' is the first and f'' is the second derivative of f.
(a) Find the null space N(T) of T. What is the dimension of N(T).
(b) Find the matrix representation [T]e of T in the basis below:ch
2. Let a = [2 0; 1 -3]
(a) Find the characteristic polynomial f(t) = det (A - tI).
(b) Find the eigenvalues and corresponding eigen vectors of A.
(c) Is A diagonalizable? Justify your answer by citing a proper theorem.
(d) Find an invertible matrix Q such that D = Q-1AQ is diagonal.
3. Let w1 = [-1; 1; 0] and w2 = [0; 1; 2], w1, w2 E R^3.
(a) Use the Gram-Schimdt process to orthogonalize, and then to orthonormalize, the vectors w1, w2.
(b) Let W = span{w1, w2}. Find the orthogonal projection of y = [0; 1; 0] to W.
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Solution Summary
The solution explains how to find the matrix representation of a linear transformation, how to find eigenvalues and eigenvectors, diagonalizing a matrix, Finding orthogonal basis and how to find orthogonal projection onto a vector space.
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