1-/5 i-y. A-3i. b[ - O o. x-f -' o -^ ^ Math 545 Final Exam Fall 2011

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1 Math 545 Final Exam Fall 2011 Name: Aluf XCWl/ Solve five of the following six problems. Show all your work and justify all your answers. Please indicate below which problem is not to be graded (otherwise, if you work on all of them, then problem 6 will not be graded). Please do not grade problem (22 points) Set A := ( (a) Show that the characteristic polynomial of A is equal to (x 3) (x 2 + 3). Show your work!,. _^ p x-f -' A-3i (b) Find a basis of C 3 consisting of eigenvectors of A. Hint: Use the notation 7] = "i+vst, fj = ~ l ~^1 an d no^e th- 3^ ^^7 = 1 arid 7j 3 = 1 (so r\ rj 2 }, ~~J\ -^ U \ i_^-^ V, o -^ ^ I 1 b[ - O o 1-/5 i-y

2 (c) Find an invertible matrix P and a diagonal matrix D, both in M 3 (C), such that P~ 1 AP = D. 1 V/l <-

$O O * J A-l X ffe (b) Find a basis for each eigenspace of A. \ 3 \ \ / \ o ^ o -Jk O \~\Q.$

3 2. (22 points) Let A = and work over the field M of real numbers. a) Show that the characteristic polynomial of A is (x + l)(x I) 2. O O * J A-l X ffe (b) Find a basis for each eigenspace of A. \ 3 \ \ / \ o ^ o -Jk O \~\Q.4-1 ^-J V^oo 1 A-tl - 3 ^ - O 6> ^> 1 'I ~~( 4 1 <9 O O ] O o o (c) Check that each vector you found in part 2b is indeed an eigenvector! r (d) Find the minimal polynomial of A. Justify your answer! I

$(e) Find a basis for each V$ in the Primary Decomposition respect to A. = Vi V^ with 0 foia \0 091 (P 9f : /'i (f) Find the elementary divisors of A.$

4 (e) Find a basis for each V$ in the Primary Decomposition respect to A. = Vi V^ with 0 foia \0 091 (P 9f : /'i (f) Find the elementary divisors of A. Carefully justify your answer! ^ I/ 0 -o f So -- XX-H) ~/-\ ij 5^C- ^ to 0* uft^ww), =. -.<rvi>,^- <^Xvi, ^/^«' (g) Find the Jordan canonical form of A. Justify your answer!. I* 0 oj

5 (h) Find an invertible matrix P, such that P 1 AP is in the Jordan canonical form you provided in part 2g. Describe your method in complete sentences! Credit will not be given to a solution found by trial and error p-i a- o /O to 3 1 (a) The matrix A = satisfies P p AP = where P 0 2 ( 1 0 % Use this information to obtain formulas for the entries of the -1 1 matrix e ta as functions of t. State (in words) each algebraic property, of the exponential of a matrix, you use.

$(22 points) Let A = / 2 0 0 0 \ 1 2 0 0 0-120 o o o 2 and B = / 2 0 0 0 \ 0 2 1 0 0 0 2 1 V 0 0 0 2 ) (a) Let T : R 4 > R 4 be the linear transformation given by T(v) direct summands appear in the$

6 (b) Use yom work in part 3a to show that the solution (yi(t), yi(t}} of the system satisfying -yi(o) = a and y 2 (0) = 6 is * =.-2/1 + 2/i (t) - ae 2t + (a, - 7 v-t 4. (22 points) Let A = / \ o o o 2 and B = / \ V ) (a) Let T : R 4 > R 4 be the linear transformation given by T(v) direct summands appear in the Primary Decomposition of T? Justify your answer! / v A' ^ cx Av. How many with respect to (b) Show that for every vector v in R 4, the order m v (x) of v with respect to T is a power of (x 2). 0 O--KM, 'v/ktck

7 (c) Find the orders m ei (x], m 2 (x), m 3 (x), m 4 (x) with respect to T, for the elements of the standard basis of R 4. Hint: You may want to use the following equality (you do not need to prove it) span{v, Av, A 2 v,...} = span{u, (A 2/)u, (A 2/) 2 u,...}. IR W = / '' J (d) Use your work in part 4c in order to find a decomposition of R 4 as a direct sum (vi) (^2) (vk) of cyclic subspaces with respect to T, such that m Vi (x) is a power of a prime polynomial in R[x]. Justify your answer!.jl '(e) Are the matrices A and B similar? Use your work above to justify your answer. B is given at the beginning of Question 4. J

$Show that the complex conjugate A = a bi is an eigenvalue of T as well. hot] <^L A- h'ml A- h*a -steal A j^ o^-co^p^ /? </ *V x, 5^^ - hu) -=-\n.$

8 5. (22points) (a) Let A be an n x n matrix with real entries and T : C n > C n the linear transformation given by multiplication by A. Assume that A = a + bi is an eigenvalue of T. Show that the complex conjugate A = a bi is an eigenvalue of T as well. hot] <^L A- h'ml A- h*a -steal A j^ o^-co^p^ /? </ *V x, 5^^ - hu) -=-\n.c\) (b) Let V be an inner product space (over R) and T : V > V" an orthogonal transformation. Show that if A is an eigenvalue of T, then A=lorA = 1. A So (c) Assume that in part 5b the dimension of V is odd. Show that T has an with eigenvalue 1 or 1. Hint: Use part 5a. It U MI. n tx) MX) \ " J ji -U.

9 (d) Consider R 3 as an inner product space with respect to the dot product and let T : R 3 > R 3 be an orthogonal transformation. Assume that u is an eigenvector of T and let W be the plane orthogonal to u. Show that W is T- invariant (i.e., that T(w] belongs W, for all w in W}. e W/ IW ^ T( ftrt (e) Keep the notation of part 5d. Show that the restriction TW to W is an orthogonal transformation. (f) Keep the notation of part 5d. Assume, in addition, that the eigenvalue of u is 1 and that det(t) = 1. Show that there exists a basis /3 2 := {v,w} of W, such that the matrix of T with respect to the basis $ := {u, f, iu} is of the /I 0 0 \ form 0 cos($) sin((9), for some angle $. Hint: You may use the fact \ 0 sin(^) cos(0) / that a 2 x 2 orthogonal matrix with determinant 1 is the matrix of a rotation f /? ( ho J &\. Oh ^ 1: 0 0 I O D \ L y-j r -r 1 7 : &' ^>1 ^ 'W-'

$- - ^n 5c> \ \J J J SI, Mj kj o^ yfio&jjj <5f Aj/wzJ) If ^ «/) ^r\ (j^h!$

10 r 6. (22 point ^J Pv (a) Recall that a linear transformation E : V > V is eidempotent, if E is nonzero, and E 2 = E. Show that every eidempotent linear transformation is diagonalizable. - - ^n 5c> \ \J J J SI, Mj kj o^ yfio&jjj <5f Aj/wzJ) If ^ «/) ^r\ (j^h!^ct ^^WJU^ Mj>{ A/ ^/^^^ ^7 - (b) Let T [ 3 > R 3 be a hnear transformation with standard matrix A and minimal polynomial m(:c) = (x 2) 2 (x 3). Set q\(x] = (x 3) and g 2 (x) = (x 2) 2. Find a polynomial 0,1(2) = ax + b of degree 1 and a constant polynomial a 2 (x) = c, such that ai(x)gi(o;) + a 2 (x)g2(^) 1 (the constant polynomial 1). f o 1 oi/x/io'x-f -/ and EI := a 2 (A)qz(A). I, where I is the identity matrix. how that 10

(d) Keep the notation of part 6c. Show that EiE 2 = 0. e) Keep the notation of part 6c. Show that EI and E% are idempotent matrices. S o <So -~. yfe j - 1 0 1 (f) Let A = 1 3 1.

11 (d) Keep the notation of part 6c. Show that EiE 2 = 0. e) Keep the notation of part 6c. Show that EI and E% are idempotent matrices. S o <So -~. yfe j (f) Let A = The minimal polynomial of A is x) = (x 2) 2 (x 3). You are not asked to prove it. Calculate the matrices and EZ for this matrix A. Hint: Start with EI to save calculations. I-L o j. I i- f -/ V o o o\ o 11

ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA

ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA Kent State University Department of Mathematical Sciences Compiled and Maintained by Donald L. White Version: August 29, 2017 CONTENTS LINEAR ALGEBRA AND