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Eigenvectors Notes

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This is an extract of our Eigenvectors document, which we sell as part of our Linear Algebra II Notes collection written by the top tier of Nanyang Technological University students.

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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.

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