Eigenvectors Notes

Linear Algebra II, Eigenvectors and values. Author: Andre Sostar Exercise: T is the linear operator on P2 (R), defined by T (f (x)) = f (x) + (x + 1)f ' (x), b is the standard ordered basis for P2 (R), and A = [T ]b . Solution: We have T : P2 (R) == P2 (R), with b = {1, x, x2 } and dim = 3. Then, we have to find A = [T ]b = T @ b in terms of g. T (1) = 1 + (x + 1) * 0 == 1 * 1 + 0 * x + *x2 = (1,0,0) ' T (x) = x + (x + 1) * (x) = 2x + 1 == 1 * 1 + 2 * x + 0 * x2 = (1,2,0) 2 2 2 ' 2 2 T (x ) = x + (x + 1) * (x ) = x + 2x(x + 1) = 3x + 2x == 0 * 1 + 2 * x + 3 * x2 = (0,2,3) Then, we transpose the result and get the answer? 1 1 0 A = [T ]b = ?0 2 2? 0 0 3 Now, we have to find the characteristic polynomial? 1-a 1 0 2-a 2 ? = (1 - a)(2 - a)(3 - a) det(A - I3) = det ? 0 0 0 3-a Therefore we can see that:? ?a1 = 1 a2 = 2? a3 = 3 Hence is an eigenvalue of T iff = 1, 2, 3. 1

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