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Course: MATH 235/237, Spring 2010
School: Waterloo
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235 Assignment Math 3 Due: Wednesday, May 26th 1. For each of the following pairs of vector spaces, dene an explicit isomorphism to establish that the spaces are isomorphic. Prove that your map is an isomorphism. a) P3 and R4 . b) The vector space P = {p(x) P3 | p(1) = 0} and the vector space U of 2 2 upper triangular matrices. 2. Suppose that {v1 , . . . , vr } is a linearly independent set in a vector space...

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235 Assignment Math 3 Due: Wednesday, May 26th 1. For each of the following pairs of vector spaces, dene an explicit isomorphism to establish that the spaces are isomorphic. Prove that your map is an isomorphism. a) P3 and R4 . b) The vector space P = {p(x) P3 | p(1) = 0} and the vector space U of 2 2 upper triangular matrices. 2. Suppose that {v1 , . . . , vr } is a linearly independent set in a vector space V and that L : V W is a one-to-one linear map. Prove {L(v1 that ), . . . , L(vr )} is a linearly independent set in W . 3. Let L be a linear operator on an n dimensional vector space V . Prove that the following are equivalent. 1) L1 exists. 2) L is one-to-one. 3) null(L) = {0} 4) L is onto. 4. Let V be an n dimensional vector space with basis B and let S be the vector space of all linear operators L : V V . Dene T : S M (n, n) by T (L) = [L]B . a) Prove that T is an isomorphism. b) Use a) to nd a basis for S .
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Waterloo - MATH - 235/237
Math 235Assignment 3 Solutions1. For each of the following pairs of vector spaces, dene an explicit isomorphism to establish that the spaces are isomorphic. Prove that your map is an isomorphism. a) P3 and R4 . Solution: We dene L : P3 R4 by L(a3 x3 + a
Waterloo - MATH - 235/237
Math 235Assignment 4Due: Wednesday, Jun 2nd1. Prove that the product of two orthogonal matrices is an orthogonal matrix. 2. Prove that if R is an orthogonal matrix, then det R = 1. Give an example of a matrix A that has det A = 1, but is not orthogonal
Waterloo - MATH - 235/237
Math 235Assignment 4 Solutions1. Prove that the product of two orthogonal matrices is an orthogonal matrix. Solution: Let P and Q be orthogonal matrices. Then we have (P Q)T (P Q) = QT P T P Q = QT Q = I, since P T P = I and QT Q = I . Thus P Q is also
Waterloo - MATH - 235/237
Waterloo - MATH - 235/237
Math 235Assignment 5 Solutionsa) Use the Gram-Schmidt process to produce an orthonormal basis for S . 2 1 1 0 1 0 Solution: Denote the given basis by z1 = , z2 = , z3 = . Let w1 = z1 . 1 1 1 1 1 1, 1 2 1 1 1 0 1 3 z2 w1 Then, we get w2 = z2 projw1 (z2 )
Waterloo - MATH - 235/237
Waterloo - MATH - 235/237
Math 235Assignment 6 Solutions1. Show that the following are equivalent for a symmetric matrix A: (1) A is orthogonal (2) A2 = I (3) All the eigenvalues of A are 1 Solution: (1) (2) (2) (3) If A is orthogonal then I = AAT = AA, since A is symmetric. Av
Waterloo - MATH - 235/237
Math 235Assignment 7Due: Wednesday, June 30th1. For each quadratic form Q(x), determine the corresponding symmetric matrix A. By diagonalizing A, Write Q so that it has no cross terms and give the change of variables which brings it into this form. Cla
Waterloo - MATH - 235/237
Math 235Assignment 7 Solutions1. For each quadratic form Q(x), determine the corresponding symmetric matrix A. By diagonalizing A, Write Q so that it has no cross terms and give the change of variables which brings it into this form. Classify each quadr
Waterloo - MATH - 235/237
Waterloo - MATH - 235/237
Math 235Assignment 8Due: Wednesday, July 14th1. Sketch the graph of 9x2 + 4xy + 6y 2 = 21 showing both the original and new axes. Solution: The corresponding symmetric matrix is polynomial is C ( ) = 9 2 = 2 15 + 50 = ( 10)( 5). 2 6 A 1 I = 1 2 1 2 . 2
Waterloo - MATH - 235/237
Math 235Assignment 9Due: Wednesday, July 21st1. Suppose that a real 2 2 matrix A has 2 + i as an eigenvalue with a corresponding 1+i eigenvector . Determine A. i 0 2 1 2. Determine a real canonical form of A = 2 2 1 and give a change of basis matrix 0
Waterloo - MATH - 235/237
Math 235Assignment 9 Solutions1. Suppose that a real 2 2 matrix A has 2 + i as an eigenvalue with a corresponding 1+i eigenvector . Determine A. i Solution: Since A is real, we know that A has real canonical form B = brought into this form by P = 11 . W
Waterloo - MATH - 235/237
Math 235Assignment 10 Not To Be Submitted 1+i 1i 1. Consider C3 with its standard inner product. Let z = 2 i , w = 2 3i. 1 + i 1 a) Evaluate z , w and w, 2iz . b) Find a vector in spancfw_z, w that is orthogonal to z . c) Write the formula for the proj
Waterloo - MATH - 235/237
Math 235Assignment 10 Solutions 1+i 1i 1. Consider C3 with its standard inner product. Let z = 2 i , w = 2 3i. 1 + i 1 a) Evaluate z , w and w, 2iz . Solution: We have z , w = (1 + i)(1 + i) + (2 i)(2 + 3i) + (1 + i)(1) = 2i 1 + 8i + 1 i = 9i w, 2iz =
Waterloo - MATH - 235/237
Math 235 - Final Exam Fall 2009NOTE: The questions on this exam does not exactly reect which questions will be on this terms exam. That is, some questions asked on this exam may not be asked on our exam and there may be some questions on our exam not ask
Waterloo - MATH - 235/237
Math 235Final F09 AnswersNOTE: These are only answers to the problems and not full solutions! On the nal exam you will be expected to show all steps used to obtain your answer. 1. Short Answer Problems 3 i i a) A = . 2 1 b) A is Hermitian since A = A, a
Waterloo - MATH - 235/237
Math 235 - Final Exam Spring 2009NOTE: The questions on this exam does not exactly reect which questions will be on this terms exam. That is, some questions asked on this exam may not be asked on our exam and there may be some questions on our exam not a
Waterloo - MATH - 235/237
Math 235Final S09 AnswersNOTE: These are only answers to the problems and not full solutions! On the nal exam you will be expected to show all steps used to obtain your answer. 1. a) A basis for the nullspace is cfw_x, hence the nullity of L is 1. Thus,
Waterloo - MATH - 235/237
Math 235Final Exam InformationThursday August 5, 9:00 AM - 11:30 AMLOCATION: PAC 1, 2, 3Material Covered: Entire Course, with an emphasis on material after term test 2. Information: - Surfaces in R3 are not covered. - Fourier Series are not covered. -
Waterloo - MATH - 235/237
Math 235 1. Short Answer ProblemsSample Term Test 1 - 1a) Give the denition of an inner product , on a vector space V . b) Let B = cfw_v1 , . . . , vn be orthonormal in an inner product space V and let v V such that v = a1 v1 + + an vn . Prove that ai
Waterloo - MATH - 235/237
Math 235Sample Term Test 1 - 1 AnswersNOTE: - Only answers are provided here (and some proofs). On the test you must provide full and complete solutions to receive full marks. 1. Short Answer Problems a) Give the denition of an inner product , on a vect
Waterloo - MATH - 235/237
Math 235 1. Short Answer ProblemsSample Term Test 1 - 2 1 0 0 1 a) Write a basis for the rowspace, columnspace and nullspace of A = 0 0 1 1 . 000 0 b) Let B = cfw_v1 , . . . , vn be orthonormal in an inner product space V and let v = a1 v1 + + an vn .
Waterloo - MATH - 235/237
Math 235Sample Term Test 1 - 2 AnswersNOTE: - Only answers are provided here (and some proofs). On the test you must provide full and complete solutions to receive full marks. 1. Short Answer Problems 1 0 0 1 a) Let A = 0 0 1 1 . Write a basis for the R
Waterloo - MATH - 235/237
Math 235 1. Short Answer ProblemsSample Term Test 2 - 1a) Let S be a subspace of an inner product space V . What is the denition of S . b) State the Principal Axis Theorem. c) Determine the matrix for the quadratic form Q(x, y, z ) = 3x2 y 2 + z 2 2xy +
Waterloo - MATH - 235/237
Math 235Sample Term Test 2 - 1 AnswersNOTE: - Only answers are provided here (and some proofs). On the test you must provide full and complete solutions to receive full marks. 1. Short Answer Problems a) Let S be a subspace of an inner product space V .
Waterloo - MATH - 235/237
Math 235 1. Short Answer ProblemsSample Term Test 2 - 2a) State the Principal Axis Theorem. b) Let A be an m n matrix. Prove that AT A is symmetric. c) State the denition of a quadratic form Q(x) on Rn being negative denite. d) Consider the quadratic fo
Waterloo - MATH - 235/237
Math 235Sample Term Test 2 - 2 AnswersNOTE: - Only answers are provided here (and some proofs). On the test you must provide full and complete solutions to receive full marks. 1. Short Answer Problems a) State the Principal Axis Theorem. Solution: A mat
Waterloo - MATH - 235/237
Math 235Midterm InformationTuesday, June 8th, 4:30 - 6:20 p.mMaterial Covered: Sections 4-5, 4-6, 4-7, 7-4 (not including Fourier series), 7-1. You need to know: - All denitions and statements of theorems. - How to nd a basis of the rowspace, column sp
Waterloo - MATH - 235/237
Math 235Term Test 2 InformationTuesday, July 6th, 4:30 - 6:20 p.mRoom Assignments: MC 4059: A - G MC 4061: H - Lin MC 4045: Liu - P MC 4020: Q - Wang MC 4021 Wardell - Z Material Covered: Sections 7-2, 7-3, Triangularization, 8-1, 8-2. You need to know
Waterloo - MATH - 235/237
Math 235Tutorial: Term Test 1 Review1: State the denition of: a) One-to-one b) Onto c) An orthogonal matrix (what are 2 other equivalent denitions?) d) An inner product 1 0 1/2 1/2 1/2 1/2 , , 2 1 1/2 1/2 1/2 1/2 0 2 T under the inner product A, B = tr
Waterloo - MATH - 235/237
SOSMATH235MIDTERM2REVIEWPACKAGE HelloMATH235students,mynameisTaiCaiandIamtheSOStutorthistermforMATH235.This packageisdesignedtosupplementyourstudyingforthesecondmidtermonNovember16,2010. Wheneverpossible,Ihaveincludedexamplesthatarenotfromclassoryourtextb
Waterloo - MATH - 235/237
Math 235 1. Short Answer ProblemsTerm Test 1 Solutions[1] a) State the denition of the rank of a linear mapping L : V W . Solution: rank(L) = dim Range(L).[2] b) Let B = cfw_v1 , . . . , vn be a basis for a vector space V and let L : V W be an isomorp
Waterloo - MATH - 235/237
Math 235 31 5 2 1. Let A = 2 1 32 4 7 1 5 2 3 3 2Assignment 1Due: Wednesday, Sept 22nd 0 1 0 0 1 1 0 0 01 0 2 . 1 1 00 3 1 0 4 , then the RREF of A is R = 0 7 1 0a) Find rank(A) and dim(Null(A). b) Find a basis for Row(A). c) Find a basis for Null(A).
Waterloo - MATH - 235/237
Math 235 31 5 2 1. Let A = 2 1 32 4 7 1 5 2 3 3 2Assignment 1 Solutions 3 1 0 4 , then the RREF of A is R = 0 7 1 0 0 1 0 0 1 1 0 0 01 0 2 . 1 1 00a) Find rank(A) and dim(Null(A). Solution: rank(A) = 3 and dim(Null(A) = 5 3 = 2 b) Find a basis for Row(A
Waterloo - MATH - 235/237
Waterloo - MATH - 235/237
Math 235Assignment 2 Solutions1. For each of the following linear transformations, determine a geometrically natural basis B and determine the matrix of the transformation with respect to B . a) The projection proj(3,2) : R2 R2 onto the line x = t Solut
Waterloo - MATH - 235/237
Math 235Assignment 3Due: Wednesday, Oct 6th1. For each of the following pairs of vector spaces, dene an explicit isomorphism to establish that the spaces are isomorphic. Prove that your map is an isomorphism. a) M (2, 2) and P3 . b) The vector space P
Waterloo - MATH - 235/237
Math 235Assignment 3 Solutions1. For each of the following pairs of vector spaces, dene an explicit isomorphism to establish that the spaces are isomorphic. Prove that your map is an isomorphism. a) M (2, 2) and P3 . Solution: We dene L : M (2, 2) P3 by
Waterloo - MATH - 235/237
Math 235Assignment 4Due: Wednesday, Oct 13th1. Prove that the product of two orthogonal matrices is an orthogonal matrix. 2. Observe that the dot product of two vectors x, y Rn can be written as x y = xT y. Use this fact to prove that if an n n matrix
Waterloo - MATH - 235/237
Math 235Assignment 4 Solutions1. Prove that the product of two orthogonal matrices is an orthogonal matrix. Solution: Let P and Q be orthogonal matrices. Then we have (P Q)T (P Q) = QT P T P Q = QT Q = I, since P T P = I and QT Q = I . Thus P Q is also
Waterloo - MATH - 235/237
Waterloo - MATH - 235/237
Math 235Assignment 5 Solutions1. On M (2, 2) dene the inner product &lt; A, B &gt;= tr(B T A) and let S = Span 10 01 1 1 , , 01 10 01 .a) a) Use the Gram-Schmidt procedure to produce an orthonormal basis for S . 10 and 01 are already orthogonal. Then 2 1 1 v
Waterloo - MATH - 235/237
Waterloo - MATH - 235/237
Math 235 1 1. Let A = 2 4 diagonalizes A 2 2 2 andAssignment 6 Solutions 4 2. Find an orthogonal matrix P that 1 the corresponding diagonal matrix.Solution: The characteristic polynomails is 1 2 4 1 2 4 2 2 = det 2 2 2 C () = det(A I ) = det 2 4 2 1 3 0
Waterloo - MATH - 235/237
Math 235Assignment 7Due: Wednesday, Nov 10th1. For each quadratic form Q(x), determine the corresponding symmetric matrix A. By diagonalizing A, Write Q so that it has no cross terms and give the change of variables which brings it into this form. Clas
Waterloo - MATH - 235/237
Math 235Assignment 7 Solutions1. For each quadratic form Q(x), determine the corresponding symmetric matrix A. By diagonalizing A, Write Q so that it has no cross terms and give the change of variables which brings it into this form. Classify each quadr
Waterloo - MATH - 235/237
MATH237: EXAM-AID SOSNIALL W. MACGILLIVRAY (EDITOR); VINCENT CHAN (WRITER)1.October 31st, 2010 ScalarFunctions(1.1 1.2)DEFINITION 1.1. Suppose A and B are sets. A function f is a rule that determines how a subset of A is associated with a subset of B
Waterloo - MATH - 235/237
Here are answers to most of the problems. I have, of course, not included ones in which the question asks to prove or verify something. Also, for now, I have not had a chance to include pictures. Math 237 Problem Set 1 AnswersA1. a) Range: z R. b) Range:
Waterloo - MATH - 235/237
Here are answers to most of the problems. I have, of course, not included ones in which the question asks to prove or verify something. Also, for now, I have not had a chance to include pictures. Math 237 Problem Set 2 AnswersA3. a) L(a,b,c) (x, y, z ) =
Waterloo - MATH - 235/237
Here are answers to most of the problems. I have, of course, not included ones in which the question asks to prove or verify something. Also, for now, I have not had a chance to include pictures. Math 237 A1.dw (2) dtProblem Set 3 Answers = 76.A2. a) I
Waterloo - MATH - 235/237
Here are answers to most of the problems. I have, of course, not included ones in which the question asks to prove or verify something. Also, for now, I have not had a chance to include pictures. Math 237 A1. a) Hf (2, 3) = 9 6 . 6 4 Problem Set 4 Answers
Waterloo - MATH - 235/237
Math 237Problem Set 5 Answers12 , 33A1. i) (0, 1), (1, 1) and (0, 0) are all saddle points.is a local min.ii) (0, 0) is a saddle point, (1, 1/2) is a local max. iii) (0, 1/ 3) local min, (0, 1/ 3) local max, (1, 0) are saddle points. iv) (0, k ), k Z
Waterloo - MATH - 235/237
Math 237 A1. a) r = 8, =4 . 3 3 . 4Problem Set 6 Answers b) r = 2, = . 6 d) r = 5, = arctan(1/2). b) (x, y ) = (3 3/2, 3/2). d) (x, y ) = ( 3, 1). b) Area= . 4c) r = 2, = A2. a) (x, y ) = (1, 3). c) (x, y ) = (3/2, 3 3/2). A3. a) Area= . 4B1. a) Are
Waterloo - MATH - 235/237
Math 237 A1. i)Problem Set 7 Answers ii) iii)A2.A3. (4.980, 0.6224). A4. a) A5. 3e e e 3e c) 2e2 0 . 4 4 2(e 1) 2(e + 1)04 11uv 2 1 , 2 (u + v ) .A6. a) F 1 (u, v ) = ln b) DF = A7. i) ii)ex 1 1/(u v ) 1(u v ) , DF 1 = . ex 1 1/2 1/ 2 y = 2x. = k o
Waterloo - MATH - 235/237
Math 237 A2. A3. A4. 2 1 (1 1 ). 2 e 1 i) 2 (cos 1Problem Set 8 Answers 1)ii) 1 sin 1.A9. 100 units A10. A12. 3 A13. e 1. B1. 1 . 6 B2. 2(e2 + 1). B3. 4. B4. both 0 B5.5 . 241
Waterloo - MATH - 235/237
Math 237 A1. i) ii) iii) iv) v) A2.b b b 0 5 0 2 0 1 0 3 1y2 b2 2 a 1 y2 b a(1 y ) c bProblem Set 9 Answers3 3 0 2q a 1 qf (x, y, z ) dz dy dx.3 1f (x, y, z ) dz dx dy.(1 x y ) a b f (x, y, z ) dz dx dy. 0 c(1 x y ) a b tan1 (1/2) 2 2 sin f ( sin
Waterloo - MATH - 235/237
Math 237 1. Let f (x, y ) = 4 + x2 y 2.Assignment 1 Solutionsi) Sketch the domain of f and state the range of f . Solution: The domain of f is 4 + x2 y 2 0 x2 y 2 4. The range is z 0.ii) Sketch level curves and cross sections. Solution: Level Curves: k
Waterloo - MATH - 235/237
Math 237x4 y 4 x2 + y 2Assignment 2 Solutions if (x, y ) = (0, 0) Determine all points where f is continuous. if (x, y ) = (0, 0).1. Let f (x, y ) =0Solution: Since x2 + y 2 = 0 if (x, y ) = (0, 0), we have that f is continuous for all (x, y ) = (0,
Waterloo - MATH - 235/237
Math 237 1. Let f (x, y ) =xy . x2 +y 2Assignment 3 Solutionsa) Find the equation of the tangent plane of f at (1, 2, 2/5). Solution: We have fx = y (x2 + y 2 ) 2x2 y y 3 x2 y =2 (x2 + y 2 )2 (x + y 2 )2 x3 xy 2 fy = 2 (x + y 2 )2Thus, the equation of
Waterloo - MATH - 235/237
Math 237Assignment 4 Solutions1. Determine all points where the function is dierentiable. a) f (x, y ) =x3 + y 3 , x2 + y 20,if (x, y ) = (0, 0) if (x, y ) = (0, 0).4 22 3 4 22 3y y Solution: Observe that fx = x +3x +y2 )2xy and fy = y +3x +y2 )2x