5.1 please only write on paper. no explanation needed. Questions: 1,3,5,7,9, 25 on paper

5.1 Eigenvalues and Eigenvectors 299 Exercise Set 5.1 In Exercises 1-4, confirm by multiplication that x is an eigenvector of A, and find the corresponding eigenvalue. 18. (Calculus required) Let D?: C - C. be the linear opera- tor in Exercise 17. Show that if w is a positive constant, then 1. A = 3 3:* =[-41 2. A=[8 -3]: x -61 sinh vw xand cosh vexare eigenvectors of D2, and find their corresponding eigenvalues. In each part of Exercises 19-20, find the eigenvalues and the corre- sponding eigenspaces of the stated matrix operator on R2. Use geo- metric reasoning to find the answers. No computations are needed. 19. a. Reflection about the line y = x. b. Orthogonal projection onto the x-axis. c. Rotation about the origin through a positive angle of 90". In each part of Exercises 5-6, find the characteristic equation, the d. Contraction with factor k (0 S k 1). C. d. d. Expansion in the y-direction with factor k (k > 1). 6. a. 2 1 e. Shear in the y-direction by a factor k (k # 0). 1 2 2 In each part of Exercises 21-22, find the eigenvalues and the corre- c. [2 sponding eigenspaces of the stated matrix operator on R3. Use geo- NO d. _2 -7] metric reasoning to find the answers. No computations are needed. In Exercises 7-12, find the characteristic equation, the eigenvalues, 21. a. Reflection about the xy-plane. and bases for the eigenspaces of the matrix. b. Orthogonal projection onto the xz-plane. 0 1 0 c. Counterclockwise rotation about the positive x-axis 7. -2 1 0 S. 0 0 through an angle of 90. 0 1] - 2 0 d. Contraction with factor k (0 1). In Exercises 13-14, find the characteristic equation of the matrix by 23. Let A be a 2 x 2 matrix, and call a line through the origin of R inspection. invariant under A if Ax lies on the line when x does. Find -8 equations for all lines in R2, if any, that are invariant under 6 3 13. - 2 7 0 0 the given matrix. 14. 3 0 0 7 a. A = 2 7 b. A = _ !] In Exercises 15-16, find the eigenvalues and a basis for each 24. Find det(A) given that A has p(a) as its characteristic poly- eigenspace of the linear operator defined by the stated formula. nomial. [Suggestion: Work with the standard matrix for the operator.] a. p(1) = 13-212+ 1+ 5 15. T(x, y) = (x+4y, 2x + 3y) b. p(1) =24 -13 + 7 16. T (x, y, Z) = (2x - y-z, x-z, -x+ +22) [ Hint: See the proof of Theorem 5.1.4.] 17. (Calculus required) Let D2: C(-0o, co) - Cw(-00, 00) 25. Suppose that the characteristic polynomial of some matrix A be the operator that maps a function into its second is found to be p(A) = (2 - 1)(1 - 3)?(1 - 4)3. In each part, derivative. answer the question and explain your reasoning. a. Show that D2 is linear. a. What is the size of A? b. Show that if w is a positive constant, then sin vox and cos v wx are eigenvectors of D', and find their correspond- b. Is A invertible? ing eigenvalues. C. How many eigenspaces does A have