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Vector Spaces, Linear Transformations, and Eigenvalues / Eigenvectors, Basis and Diagonalizing Matrices

1. Answer the following questions
A. Is lambda = 2 an eigenvalue of the following matrix? Why or why not?
3 2
3 8
B. Is lambda - -2 an eigenvalue of the following matrix? Why or why not?
7 3
3 -1
C. Is the following vector an eigenvector of the following matrix? If so, find the eigenvalue.
1
4

-3 1
-3 8
2. Find a basis for the eigenspace corresponding to each listed eigenvalue and the following matrix
a. lambda = 1, 5
5 0
2 1
b. lambda = 4
10 -9
4 -2
3. Find the eigenvalues of the following matrices
a.
0 0 0
0 2 5
0 0 -1
b.
4 0 0
0 0 0
1 0 -3
4. In the following problem, A is an n x n (square) matrix. Mark each statement True or False. Justify your answer
a. If Ax = lambda x for some vector x, then lambda is an eigenvalue of A.
b. A matrix A is not invertible if and only if 0 is an eigenvalue of A.
c. A number c is an eigenvalue of A if and only if the equation (A - cI)x = 0 has a non-trivial solution
d. Finding an eigenvector of A may be difficult, but checking whether a given vector is in fact an eigenvector is easy.
e. To find the eigenvalues of A, reduce A to echelon form.
5. Diagonalize the following matrices if possible
a.
1 0
6 -1
b.
5 1
0 5
c.
2 3
4 1

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Vector Spaces, Linear Transformations, and Eigenvalues / Eigenvectors, Basis and Diagonalizing Matrices are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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