Math 511 Fall 2009 Final Exam - Corrected (twice)
(The change is to add the word “not” in question 3 (b).
Name
Student Number
1. Answer Tfor true or Ffor false. three points each
Assume that all vector spaces are finite dimensional and that the field in R.
(a) A linear operator on an ndimensional vector space must have nor fewer eigenvalues.
(b) A linear operator on an ndimensional vector space with at least one eigenvector will have infinitely
many eigenvectors.
(c) If vis an eigenvector of the linear operator Tcorresponding to eigenvalue 3 the T(v)−3v= 0.
(d) If Tis a linear operator on a finite dimension vector space Vand αand βare two bases for V, then
[T]αand [T]βhave the same eigenvectors.
(e) If Tis a linear operator on a finite dimension vector space Vand αand βare two bases for V, then
[T]αand [T]βhave the same eigenvalues.
(f) If Tis a linear operator on a finite dimensional vector space Vand if v1and v2are eigenvectors
corresponding to the eigenvalues λ1and λ2respectively then v1+v2is an eigenvector corresponding to
the eigenvalue λ1+λ2of T.
(g) If a linear operator Ton a vector space Vof dimension nhas at least n−1 distinct eigenvalues then
Tis diagonalizable.
(h) If λis an eigenvalue of the linear operator Tand v1and v2are eigenvectors corresponding to the
eigenvalue λ, then v1=cv2for some scalar c.
(i) If λ1is an eigenvalue of a linear operator Tthen −λis also an eigenvalue of T.
(j) If fT(t) (the characteristic polynomial of the linear operator T) doesn’t split, then Tis not diago-
nalizable.
2. Short Answer, five points each
(a) For the matrix A=
0−2−1
131
−2−4−1
i. Find the eigenvalues.
ii. Find a basis βfor R3consisting of eigenvectors of A.
iii. What is [LA]β?
(b) Why is
1 2 2
0−1−2
0 2 3
not diagonalizable. Note that λ= 1 is an eigenvalue of the matrix.
(c) Find a polynomial p(t) such that p(A) = 0 where Ais the matrix from the previous problem.
(d) Give an example of a 3 ×3 non-zero matrix Band a polynomial p(x) of degree 2 such that P(B) = 0.
(The right hand side of this equation is the zero 3 ×3 matrix.)
