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# 5.4 - 338 Chapter 5 Linear Transformations 5.4 Problems...

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Unformatted text preview: 338 Chapter 5 Linear Transformations 5.4 Problems Changing Coordinates I 1:: Problems 1—3, let S m {131,33} be the standard basis, and B=lBIwBEJ={[—:i'[wii} be a new basis for R3 1» Calculate the coordinate—change matrices M 3 to go from B to S and Mg] as in Exempie 2 l9 . Convert the vectors 3 2 0 [8i i—Ii‘ and iii from the standard basis to the basis 8 3” Vectors Lil iii and iii are expressed in basis 8, Find their standard basis representations. Changing Coordinates H In Problems 46, let S = [E], 33; be the standard basis, and 3: @332}: ' . '7 be a new bﬂ-SI‘S for 131-. 4. Caiculate the coordinate change matrices MB to go from B to S and M? as in Example 2. 5. Convert the vectors ii], iii and [3’] from the standard basis to the basis B [it Li], and [ii are expressed in basis B Find their standard basis representations. 6. Vectors Changing Coordinates III For Problems 7—9, the standard basis is S = {3:71, 353, E3}. Let Marin} be a new basis for B31? 7. Calculate the coordinate Change matrices MB to go from B to S and ME] as in Exampie 5 8. Convert the vectors lil lil lil from the standard basis to the basis Br n a n are expressed in basis 5’ Find their standard basis representations. 9. Vectors Changing Coordinates IV ForProbiemS 10—12, the standard basis is S = [é1,ég, e3}. Let B=tlélliliill be a new basisfor R3. 10. Calculate the coordinate change matrices MB to go from B to S and ME‘ as in Exampie 5 11. Convert the vectors til lily iii from the standard basis to basis Br léil l-il iii are expressed in basis B Find their standard basis representations 12. Vectors PoEynomial Coordinates I For Problems 13—15, we take 5 m {\$2, I, 1} as the standard basis in E33 and introduce a new basis N = {2x2 — x, x3, x3 + 1}‘ 13. Compute coordinate change matrices MN to go from N to S and ME] as in Example 6. 14. Express in terms of the new basis N the standard basis poiynomiais 130:) = x2 + 2.x «1—3, = x2 _ 2) r(x) = 4x — 5. Section 5.4 Coordinates and Diagonaiization 15. Vector representations of three polynomials relative to the basis N are I —2 —~1 TIN = 0 , YIN = 2 , and {hr 2 ml . .7. 3 0 Calculate standard representations of n(x}, u(.t), anti w(x}. Polynomial Coordinates II For Problems 16-18, we take S m [2:3, x3, .x, i} as the standard basis in [Pg and introduce a new basis Q = {1:3, 3:3 + x3, x2 + 1}. 16. Compute coordinate change matrices Mg to go from Q to S and M? as in Exampie 6. 17. Find the coordinate vectors of these standard basis poly— nomials in terms of the new basis Q: pix) = x3 + 2x2 + 3, qix) m .I2 wx w 2,. r'(.r) = 1'3 +1. 13. Vecto: representatioas of three polynomiais relative to the basis Q are 1 —2 3 _ wt _ 0 .. _1 HQ = 0 , vQ = _,) , and wQ m 4 2 0 2 Calculate standard representations of' n(x), nix), and w(x) Matrix Representations for Polynomiai Transformations Using the standard basis 5 = (:4, t3, t2, 1‘, I] for P4, deter‘v mine a matrix representing the transformation from P4 to Ill"; given in each ofProblem-s 19"?! Then apply your matrix to the following polynomials .' (a) g(t)=t"l w—t3 +t2—t+I (b) q(t) = r‘ + 2:1 + 4 (c) an = watt“ + 39 (ca w(t) = :4 — at1 + t6 19. mm) = f”(t) 20. Nita) = Ito) 21. T{f(t})=f’”(t) 22. turn): ft—t) 23. Two): f’(o—2f{t} 24. Win): f”(t)+f‘(!) Biagonalization 1n Problems 2548, determine whether each matrix A is diagonalizable If it is, determine a matrix P that diagonalize-S it and compute P“1AP. You can obtain P’iAP directly from careful construction ofa diagonal matrix with eigenvalues along the diagonal in the proper order. 3 2 l —1 1 2 7, . 7 -3. { 2 3] 26. [i 3] -7. [2 1] 28. 31. 34. 37. 39. 41. 4.3. 47. 49. 339 [:3] ma so-H it: :a a 1a sari-:1 3 ml i 0 0 1 7 --5 i 40. O i 2 6 we 2 O O l i 1 1 4 2 3 0 0 1 42. 2 1 2 0 O 1 —1 2 O 1 0 0 3 —~2 0 -4 .3 {3 44. l O O ~41 '2 1 ml 1 3 7 1 8 —1 O 0 2 " -1 1 2 46. 0 4 G 0 _1 0 3 0 0 6 0 O O 0 4 4 O 4i 0 2 O i 2 0 4 O O 48 0 2 O O O i} 8 O ' O O 6 0 «1 —2 1 8 O 0 1 4 Powers of 3 Matrix Suppose that A is a diagonaiizable matrix that has been written in the form A m PDP”'. where D is diagonal (a) Show that for positive integer k, Ak = PD"P" . (b) Use the result of part (a) to compute A50 for 11 A_[41l' (c) Show that if D is diagonal, then D‘L’ is diagonal (d) is the equation in part (a) true fork m --« 1‘? Could this be usef'ui in ﬁnding the inverse of a matrix? . Determinants and Eigenvaiues Let A be an n x a matrix such that its characteristic polynomial is lA — MI =01 ~ 1:)(1- h lzl- ' ' (it "" in}. where the Pt,- ar‘e distinct. (:1) Explain why m Ail-2' ")Vn- (b) If the A,- at‘e not distinct but A is diagonaiizable, would the same property hold? Explain. 340 Chapter 5 Linear Transformations Constructing Counterexamples In Problems 51m53, Con- struct the required examples. 51h Construct a 2 x 2 matrix that is invertible but not diago~ naiizabte. 01 In) . Construct a 2 x 2 matrix that is diagonalizable but not invertible 53. Construct a 2 x 2 matrix that is neither invertibie nor diagonalizable 54. Competer Lab: Diagonalization Use appropriate corn~ pater software to diagonaiize (if possibte) the following matrices till 0111 (a) 0 011 0001 m2 1 100 i—Z 100 (b) 1 1-200 0 0 011 0 0 041 3000 0110 (C) 0110 0005 55” Similar Matrices We have deﬁned matrix B to be similar to matrix A (denoted by B ~ A) if there is an invertible matrix P such that B = P‘IAP, Provo the following: (:1) Similar matrices have the same characteristic polyno- mial and the same eigenvalues. (b) Similar matrices have the same determinant and the same trace. (c) Show by exampie (using 2 x 2 matrices) that similar matrices can have different eigenvectors‘ 56. How Similar Are They? Let [4 ——2 "310 i 1] and Bm[ ] —38 (a) Show that A w B (b) Verify that A and B share the properties discussed in parts (a) and (h) of Problem 55. 57. Computer Lab: Similarity Challenge Repeat Prob— lem 55 for the more challenging case of two 3 x 3 matrices. using a computer aigebra system 1 2 —3 l w19 58 Am 2 0 1 and B: 1 12 —27 t l —«3 1 5 15 ~11 58. Orthogonal Matrices An orthogonai matrix P is a square matrix whose transpose equals its inverse: Pi mP-tt (a) Show that this is equivalent to the condition PP1r = 1.. (13) Use part (a) to show that the column vectors of an orthogonal matrix are orthogonal vectors. (See Sec 3 i ) 59. Orthogonally Diagonalizable Matrices A matrix A is orthogonally diagonalizahie if there is an orthogonal ma- trix P that diagonalizes it. Show that the matrix 4 2 A ‘ [2 7] is orthogonaiiy diagonalizable, HtNT: Symmetric matrices have orthogonal eigenvectors. 60. When Diagonalization Fails Prove that, for a 2 x 2 ma— trix A with a double eigenvalue but a single eigenvector a z[v1,vg],v3 # 0, the matrix Q: and its inverse Q“1 can provide a change of basis fOr A, such that Q‘IAQ is a triangutar matrix (which wilt have the eigenvalues on the diagonal, as shown in Sec. 5 3, Probiems 43-46), Triangularizing Apply the procedure of Problem 60 to triam gularize the matrices ofProblems 61—62 2 —t i 1 ti 5] t1 4.] 6.3“ Suggested Journal Entry I Discuss how you might go about deﬁning coordinates for a vector in an inﬁnite- dimensional vector Space, C[0, l}, for example. Would it make a difference if you just wanted to be able to approx— imate vectors rather than obtaining exact representations? 64. Suggested Journal Entry 11 Explain and elaborate the assertion by Gilbert Strung of the need to emphasize that “Diagonalizabiiity is concerned with the eigenvectors‘ Inw vertibility is concerned with the eigenvalues."3 3Gi1bert Strung. Linear Algebra and Its Applications, 3rd edition (Harcourt Brace iovanovich, 1988). 256 Section 5.4 E Coordinates and Siagonaiizatéoo .34] IA 1-, m, J I J I r r T r r r r I l r (A) (B) F I G U R E 5 .4 n 6 Phase portrait and solution gtaphs in try coordinates for Problem 65 65. Suggested Journal Entry III Figure 5 4.6(A) shows the phase portrait of Example 7, l“l'- l1 1l H y _ 4 1 y ‘ (a) Add to the phase portrait the eigenvectors [1,2] and [1, will}, which go with the eigenvaiues 3 and _1, respectively {13) Why might we call the eigenvectors “nature’s coordiw nate system," and the original xy coordinates “human laboratory coordinates”? (c) (d) Figure 5.4603) and (C) Show solution graphs for .150) and ﬁt); what sort of solution graphs would you expect if new coordinate axes are aligned along the eigenvector‘s? Relate your discoveries in parts (b) and (c) to the fol— lowing support statement for diagonaiization: “One of the goals of all science is to ﬁnd the simplest way to describe a physical system, and eigenvectors just happen to be the way to do it for linear systems of differential equations” ...
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5.4 - 338 Chapter 5 Linear Transformations 5.4 Problems...

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