Problems : 9,11,24,25,26 Direction : in the exercices 9 and 11 find a basis for the eigenspace corresponding to each listed eigenvalueHello can I have some help here please. Can you write on a sheet of paper an upload it please Reference :section 5.1 David C. Lay, Steven R. Lay, Judi J. McDonald - Linear Algebra and Its Applications-Pearson (2015).pdf

9. A = 2 9 . 2 = 1,5 d. Finding an eigenvector of A may be difficult, but check- ing whether a given vector is in fact an eigenvector is easy. 10. A = 4 2 .1 = 4 e. To find the eigenvalues of A, reduce A to echelon form. 22. a. If Ax = Ax for some scalar 1, then x is an eigenvector 11. A = 3 3].1 = 10 of A. b. If vi and v2 are linearly independent eigenvectors, then they correspond to distinct eigenvalues. [-3 _1].A - 15 . A steady-state vector for a stochastic matrix is actually an eigenvector. d. The eigenvalues of a matrix are on its main diagonal. 13. A = -2 1 0 , 1 = 1,2 ,3 e. An eigenspace of A is a null space of a certain matrix. 23. Explain why a 2 x 2 matrix can have at most two distinct eigenvalues. Explain why an n x n matrix can have at most n distinct eigenvalues. 14. A = 24. Construct an example of a 2 x 2 matrix with only one distinct eigenvalue. 25. Let A be an eigenvalue of an invertible matrix A. Show that 15. A = 2 I , 1 = 3 -is an eigenvalue of A . [Hint: Suppose a nonzero x satisfies Ax = Ax.] 26. Show that if A2 is the zero matrix, then the only eigenvalue 2 of A is 0. 16. A = 1 27. Show that A is an eigenvalue of A if and only if a is an , 1 = 4 eigenvalue of AT . [Hint: Find out how A - 1/ and AT - 1I 0 0 0 are related.] Find the eigenvalues of the matrices in Exercises 17 and 18. 28. Use Exercise 27 to complete the proof of Theorem 1 for the case when A is lower triangular. 0 0 29. Consider an n x n matrix A with the property that the row 17. oo 2 0 sums all equal the same number s. Show that s is an eigen- value of A. [Hint: Find an eigenvector.] 30. Consider an n x n matrix A with the property that the col- 19. For A = 1 2 3 , find one eigenvalue, with no cal- umn sums all equal the same number s. Show that s is an eigenvalue of A. [Hint: Use Exercises 27 and 29.] culation. Justify your answer. In Exercises 31 and 32, let A be the matrix of the linear transfor- mation 7 . Without writing A, find an eigenvalue of A and describe 20. Without calculation, find one eigenvalue and two linearly the eigenspace. 5 31. T is the transformation on R2 that reflects points across some independent eigenvectors of A = 5 Justify line through the origin. your answer. 32. T is the transformation on R' that rotates points about some line through the origin. In Exercises 21 and 22, A is ann x n matrix. Mark each statement 33. Let u and v be eigenvectors of a matrix A, with corresponding True or False. Justify each answer. eigenvalues 1 and M, and let c, and c2 be scalars. Define 21. a. If Ax = Ax for some vector x, then ) is an eigenvalue of Kk = cidku+ czukv (k = 0, 1, 2, ...) A . a. What is Xk+1, by definition? b. A matrix A is not invertible if and only if 0 is an eigen- b. Compute Axk from the formula for Xk, and show that value of A. AXk = Xk+1. This calculation will prove that the se- c. A number c is an eigenvalue of A if and only if the quence {xx } defined above satisfies the difference equa- equation (A - cl )x = 0 has a nontrivial solution. tion Xk +1 = AXk (k = 0, 1, 2, ...)