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## Eigenvectors Notes

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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