Mathematics 206 Solutions for HWK 23 Section 6.3 p358
|
|
- Logan Fox
- 5 years ago
- Views:
Transcription
1 Mathematics 6 Solutions for HWK Section Problem 9. Given T(x, y, z) = (x 9y + z,6x + 5y z) and v = (,,), use the standard matrix for the linear transformation T to find the image of the vector v. Note that the domain for T is R and the codomain is R. So we re expecting a matrix. Since T(,, ) = (, 6), T(,,) = ( 9,5), and T(,, ) = (, ), the standard matrix for T is ] 9 A = 6 5 and the standard coordinate matrix for T(v) is A In other words, T(v) = (5, 7). = ] = ] 5. 7 Problem. Given that T : R R is the reflection through the origin, T(x, y) = ( x, y), and given v = (, ), (a) find the standard matrix A for the linear transformation T, (b) use A to find the image of the vector v, and (c) sketch the graph of v and its image. (a) Since T(, ) = (, ) and T(, ) = (, ), the standard matrix is A = ]. (b) T(v)] = Av] = ] ] = ] (c) See the sketch in the text. Page of 8 A. Sontag May,
2 Math 6 HWK Solns contd Problem 5. Given that T : R R is the counterclockwise rotation of 5 in R, and given v = (, ), (a) find the standard matrix A for the linear transformation T, (b) use A to find the image of the vector v, and (c) sketch the graph of v and its image. (a) T(,) = (cos 5,sin 5 ) = (, ) T(,) = (cos 5,sin 5 ) = (, ) A = ] (b) T(v)] = A v] = ] ] = ] T(v) = (, ) (c) See the text for a sketch. Problem 7. Given that T : R R is the reflection through the xy-coordinate plane in R, T(x, y, z) = (x, y, z), and given that v = (,, ), (a) find the standard matrix A for the linear transformation T, (b) use A to find the image of the vector v, and (c) sketch the graph of v and its image. (a) T(,,) = (,,), T(,,) = (,, ), and T(,,) = (,, ) so A = (b) T(v)] = Av = = so T(v) = (,, ).. (c) See the text for a sketch. Page of 8 A. Sontag May,
3 Math 6 HWK Solns contd Problem. Given that T : R R is the projection onto the vector w = (,) in R, T(v) = projwv, and given that v = (, ), (a) find the standard matrix A for the linear transformation T, (b) use A to find the image of the vector v, and (c) sketch the graph of v and its image. (a) (b) T(v)] = Av = T(, ) = proj (,) (,) = (,) T(, ) = proj (,) (,) = (,) A = 9 ] 9 ] ] = T(v) = 7 (,) ] = 7 7 ] (c) See the sketch in the text. Problem 5. Find the standard matrices for T = T T and T = T T, given T : R R, T (x, y) = (x y, x + y) T : R R, T (x, y) = (x, x y) ] The standard matrix for T is A =. ] The standard matrix for T is A =. In this situation matrix multiplication corresponds ] to composition of functions, so the standard matrix for T is A A = 5 ] and the standard matrix for T is A A =. 7 Page of 8 A. Sontag May,
4 Math 6 HWK Solns contd Problem 7. Given T : R R, T(x, y) = (x + y, x, y), v = (5, ) B = {(, ),(, )}, B = {(,,),(,,), (,, )} find T(v) by using (a) the standard matrix and (b) the matrix relative to B and B. (a) The standard matrix for T is A = so and T(v) = (9, 5,). ] 9 T(v)] = Av] = 5 = 5 (b) Denote the vectors in B, in the order given, as v, v. Similarly, let the vectors in B be called w, w, w. Then T(v ) = (,, ) = w w, T(v ) = (,, ) = w. Therefore the matrix for T, relative to B and B is A = Moreover, v = 5v + 9v. This gives ] T(v )] B T(v )] B =. T(v)] B = A v] B = ] 5 9 = 5 and consequently T(v) = 5w + w = (9,5, ), which agrees with the result found in (a). Page of 8 A. Sontag May,
5 Math 6 HWK Solns contd Problem. Given T : R R, T(x, y, z) = (x, x + y, y + z, x + z), v = (, 5,) B = {(,, ),(,, ),(,,)}, B = {(,,,),(,,, ),(,,,), (,,,)} find T(v) by using (a) the standard matrix and (b) the matrix relative to B and B. (a) The standard matrix for T is A = so so T(v) = (,,,). T(v)] = Av] = 5 = (b) Denote the vectors in B by v, v, v, and those in B by w, w, w, w. Then T(v ) = (,,, ) = w + w + w + w T(v ) = (,,, ) = w + w + w w T(v ) = (,,, ) = w + w + w + w The matrix for T relative to B and B is therefore A = Moreover, v = 9 v + v 8v. Therefore ] T(v )] B T(v )] B T(v )] B = T(v)] B = A v] B = = which gives T(v) = 6w w w w = (,,,), which agrees with the result from (a). Page 5 of 8 A. Sontag May,
6 Math 6 HWK Solns contd Problem 5. Let T : P P be given by T(p) = xp. (In other words, if p(x) = c + c x + c x, then T(p) is the polynomial defined by (T(p))(x) = x(p(x)) = c x + c x + c x. Find the matrix of T relative to the bases B = {, x, x } and B = {, x, x, x }. Therefore the required matrix is T() = x, so T()] B =. T(x) = x, so T(x)] B =. T(x ) = x, so T(x )] B =. ] T()] B T(x)] B T(x )] B =. Problem 7. Let B = {, x, e x, xe x } be a basis of a subspace W of the space of continuous functions, and let D x be the differential operator on W. (In other words D x : W W is the linear transformation defined by D x (f) = f = the derivative function for the function f.) Find the matrix for D x relative to the basis B. D x () =, D x (x) =, D x (e x ) = e x, and D x (xe x ) = e x + xe x, so the required matrix is ] A = D x ()] B D x (x)] B D x (e x )] B D x (xe x )] B = Page 6 of 8 A. Sontag May,
7 Math 6 HWK Solns contd Problem 9. Use the matrix from Exercise 7 to evaluate D x (x xe x ). Let f be the function we wish to differentiate using Exercise 7. Then D x (x xe x )] B = D x (f)] B = Af] B = = This tells us that D x (f)(x) = e x xe x, exactly as we would expect from calculus. Problem 5. Let B = {, x, x, x } be a basis for P, and let T : P P be the linear transformation given by T(x k ) = t k dt. (a) Find the matrix A for T with respect to B and the standard basis for P. (b) Use A to integrate p(x) = 6 x + x. (a) T() = T(x ) = T(x ) = dt = x, T(x) = t dt = x, T(x ) = t dt = x t dt = x Give the name B to the standard basis for P : B = {, x, x, x, x }. Then the required matrix for T is ] A = T()] B T(x)] B T(x )] B T(x )] B = (b) The instructions are a little vague. Let s assume that what s wanted is to find T(6 x+x ). Then we have T(6 x + x )] B = A 6 6 x + x ] B = Thus T(6 x + x ) = 6x x + x. = 6. Page 7 of 8 A. Sontag May,
8 Math 6 HWK Solns contd Problem 55. Let T : M, M, be given by T(A) = A T. Find the matrix for T relative to the standard bases for M, and M,. Having consulted the text on p, and following the order suggested by Example 5, I ll take the standard basis for M, to be {M, M, M, M, M 5, M 6 }, where M, M, and M have zeros in all positions of the bottom row and all except one position of the top row and they have in the first, second, third positions, respectively of the first row. Then M, M 5, M 6 have zeros in all positions of the top row and two positions of the bottom row, with in the first, second, and third positions, respectively of the second row. (Yes, I m trying to avoid having to type in all those matrices.) I ll take the standard basis for M,, which I ll write as B = {N, N, N, N, N 5, N 6 }, to be arranged in like fashion: first let the s move across the first row, then across the second row, and finally across the third row, always from left to right. If you ordered these two bases differently, your representing matrix will come out different from mine, but your results should be consistent with mine once you take that difference into account. So here goes, finally. T(M ) = (M ) T = N, T(M ) = (M ) T = N T(M ) = (M ) T = N 5, T(M ) = (M ) T = N T(M 5 ) = (M 5 ) T = N, T(M 6 ) = (M 6 ) T = N 6 Therefore the matrix we want is. Page 8 of 8 A. Sontag May,
Mathematics 206 Solutions for HWK 22b Section 8.4 p399
Mathematics Solutions f HWK b Section 8. p99 Problem, 8. p99. Let T : P P be the linear transfmation defined by T(p(x)) = xp(x). (a) Find the matrix f T with respect to the standard bases B = {u, u, u
More informationMATH Topics in Applied Mathematics Lecture 2-6: Isomorphism. Linear independence (revisited).
MATH 311-504 Topics in Applied Mathematics Lecture 2-6: Isomorphism. Linear independence (revisited). Definition. A mapping f : V 1 V 2 is one-to-one if it maps different elements from V 1 to different
More informationMath Exam 2, October 14, 2008
Math 96 - Exam 2, October 4, 28 Name: Problem (5 points Find all solutions to the following system of linear equations, check your work: x + x 2 x 3 2x 2 2x 3 2 x x 2 + x 3 2 Solution Let s perform Gaussian
More informationMath 121 Winter 2010 Review Sheet
Math 121 Winter 2010 Review Sheet March 14, 2010 This review sheet contains a number of problems covering the material that we went over after the third midterm exam. These problems (in conjunction with
More informationAnnouncements September 19
Announcements September 19 Please complete the mid-semester CIOS survey this week The first midterm will take place during recitation a week from Friday, September 3 It covers Chapter 1, sections 1 5 and
More informationSection 1.8/1.9. Linear Transformations
Section 1.8/1.9 Linear Transformations Motivation Let A be a matrix, and consider the matrix equation b = Ax. If we vary x, we can think of this as a function of x. Many functions in real life the linear
More informationMathematics 206 Solutions for HWK 13b Section 5.2
Mathematics 206 Solutions for HWK 13b Section 5.2 Section Problem 7ac. Which of the following are linear combinations of u = (0, 2,2) and v = (1, 3, 1)? (a) (2, 2,2) (c) (0,4, 5) Solution. Solution by
More information5.) For each of the given sets of vectors, determine whether or not the set spans R 3. Give reasons for your answers.
Linear Algebra - Test File - Spring Test # For problems - consider the following system of equations. x + y - z = x + y + 4z = x + y + 6z =.) Solve the system without using your calculator..) Find the
More informationMath 212-Lecture 8. The chain rule with one independent variable
Math 212-Lecture 8 137: The multivariable chain rule The chain rule with one independent variable w = f(x, y) If the particle is moving along a curve x = x(t), y = y(t), then the values that the particle
More informationMATH 1553 SAMPLE FINAL EXAM, SPRING 2018
MATH 1553 SAMPLE FINAL EXAM, SPRING 2018 Name Circle the name of your instructor below: Fathi Jankowski Kordek Strenner Yan Please read all instructions carefully before beginning Each problem is worth
More informationDirect Sums and Invariants. Direct Sums
Math 5327 Direct Sums and Invariants Direct Sums Suppose that W and W 2 are both subspaces of a vector space V Recall from Chapter 2 that the sum W W 2 is the set "u v u " W v " W 2 # That is W W 2 is
More informationLIMITS AT INFINITY MR. VELAZQUEZ AP CALCULUS
LIMITS AT INFINITY MR. VELAZQUEZ AP CALCULUS RECALL: VERTICAL ASYMPTOTES Remember that for a rational function, vertical asymptotes occur at values of x = a which have infinite its (either positive or
More informationSTUDENT NAME: STUDENT SIGNATURE: STUDENT ID NUMBER: SECTION NUMBER RECITATION INSTRUCTOR:
MA262 FINAL EXAM SPRING 2016 MAY 2, 2016 TEST NUMBER 01 INSTRUCTIONS: 1. Do not open the exam booklet until you are instructed to do so. 2. Before you open the booklet fill in the information below and
More informationReview all the activities leading to Midterm 3. Review all the problems in the previous online homework sets (8+9+10).
MA109, Activity 34: Review (Sections 3.6+3.7+4.1+4.2+4.3) Date: Objective: Additional Assignments: To prepare for Midterm 3, make sure that you can solve the types of problems listed in Activities 33 and
More informationHomework 2 Solutions
Math 312, Spring 2014 Jerry L. Kazdan Homework 2 s 1. [Bretscher, Sec. 1.2 #44] The sketch represents a maze of one-way streets in a city. The trac volume through certain blocks during an hour has been
More informationMath 20F Final Exam(ver. c)
Name: Solutions Student ID No.: Discussion Section: Math F Final Exam(ver. c) Winter 6 Problem Score /48 /6 /7 4 /4 5 /4 6 /4 7 /7 otal / . (48 Points.) he following are rue/false questions. For this problem
More informationMath 211 Business Calculus TEST 3. Question 1. Section 2.2. Second Derivative Test.
Math 211 Business Calculus TEST 3 Question 1. Section 2.2. Second Derivative Test. p. 1/?? Math 211 Business Calculus TEST 3 Question 1. Section 2.2. Second Derivative Test. Question 2. Section 2.3. Graph
More informationAssignment 1 Math 5341 Linear Algebra Review. Give complete answers to each of the following questions. Show all of your work.
Assignment 1 Math 5341 Linear Algebra Review Give complete answers to each of the following questions Show all of your work Note: You might struggle with some of these questions, either because it has
More informationMath 261 Calculus I. Test 1 Study Guide. Name. Decide whether the limit exists. If it exists, find its value. 1) lim x 1. f(x) 2) lim x -1/2 f(x)
Math 261 Calculus I Test 1 Study Guide Name Decide whether the it exists. If it exists, find its value. 1) x 1 f(x) 2) x -1/2 f(x) Complete the table and use the result to find the indicated it. 3) If
More informationInterpolation and the Lagrange Polynomial
Interpolation and the Lagrange Polynomial MATH 375 J. Robert Buchanan Department of Mathematics Fall 2013 Introduction We often choose polynomials to approximate other classes of functions. Theorem (Weierstrass
More informationPractice Final Exam Solutions for Calculus II, Math 1502, December 5, 2013
Practice Final Exam Solutions for Calculus II, Math 5, December 5, 3 Name: Section: Name of TA: This test is to be taken without calculators and notes of any sorts. The allowed time is hours and 5 minutes.
More informationAnnouncements Wednesday, September 27
Announcements Wednesday, September 27 The midterm will be returned in recitation on Friday. You can pick it up from me in office hours before then. Keep tabs on your grades on Canvas. WeBWorK 1.7 is due
More informationMath 250B Midterm II Information Spring 2019 SOLUTIONS TO PRACTICE PROBLEMS
Math 50B Midterm II Information Spring 019 SOLUTIONS TO PRACTICE PROBLEMS Problem 1. Determine whether each set S below forms a subspace of the given vector space V. Show carefully that your answer is
More informationSolutions to Math 51 First Exam October 13, 2015
Solutions to Math First Exam October 3, 2. (8 points) (a) Find an equation for the plane in R 3 that contains both the x-axis and the point (,, 2). The equation should be of the form ax + by + cz = d.
More informationMath 233. Directional Derivatives and Gradients Basics
Math 233. Directional Derivatives and Gradients Basics Given a function f(x, y) and a unit vector u = a, b we define the directional derivative of f at (x 0, y 0 ) in the direction u by f(x 0 + ta, y 0
More information2, or x 5, 3 x 0, x 2
Pre-AP Algebra 2 Lesson 2 End Behavior and Polynomial Inequalities Objectives: Students will be able to: use a number line model to sketch polynomials that have repeated roots. use a number line model
More informationMath 221 Notes on Rolle s Theorem, The Mean Value Theorem, l Hôpital s rule, and the Taylor-Maclaurin formula. 1. Two theorems
Math 221 Notes on Rolle s Theorem, The Mean Value Theorem, l Hôpital s rule, and the Taylor-Maclaurin formula 1. Two theorems Rolle s Theorem. If a function y = f(x) is differentiable for a x b and if
More informationProcedure for Graphing Polynomial Functions
Procedure for Graphing Polynomial Functions P(x) = a nx n + a n-1x n-1 + + a 1x + a 0 To graph P(x): As an example, we will examine the following polynomial function: P(x) = 2x 3 3x 2 23x + 12 1. Determine
More informationMath 21b Final Exam Thursday, May 15, 2003 Solutions
Math 2b Final Exam Thursday, May 5, 2003 Solutions. (20 points) True or False. No justification is necessary, simply circle T or F for each statement. T F (a) If W is a subspace of R n and x is not in
More informationExam 3 MATH Calculus I
Trinity College December 03, 2015 MATH 131-01 Calculus I By signing below, you attest that you have neither given nor received help of any kind on this exam. Signature: Printed Name: Instructions: Show
More informationPreliminaries Lectures. Dr. Abdulla Eid. Department of Mathematics MATHS 101: Calculus I
Preliminaries 2 1 2 Lectures Department of Mathematics http://www.abdullaeid.net/maths101 MATHS 101: Calculus I (University of Bahrain) Prelim 1 / 35 Pre Calculus MATHS 101: Calculus MATHS 101 is all about
More information1,3. f x x f x x. Lim. Lim. Lim. Lim Lim. y 13x b b 10 b So the equation of the tangent line is y 13x
1.5 Topics: The Derivative lutions 1. Use the limit definition of derivative (the one with x in it) to find f x given f x 4x 5x 6 4 x x 5 x x 6 4x 5x 6 f x x f x f x x0 x x0 x xx x x x x x 4 5 6 4 5 6
More informationFactors of Polynomials Factoring For Experts
Factors of Polynomials SUGGESTED LEARNING STRATEGIES: Shared Reading, Activating Prior Knowledge, Discussion Group, Note-taking When you factor a polynomial, you rewrite the original polynomial as a product
More informationVector Functions & Space Curves MATH 2110Q
Vector Functions & Space Curves Vector Functions & Space Curves Vector Functions Definition A vector function or vector-valued function is a function that takes real numbers as inputs and gives vectors
More informationMath 165 Final Exam worksheet solutions
C Roettger, Fall 17 Math 165 Final Exam worksheet solutions Problem 1 Use the Fundamental Theorem of Calculus to compute f(4), where x f(t) dt = x cos(πx). Solution. From the FTC, the derivative of the
More informationSection IV.23. Factorizations of Polynomials over a Field
IV.23 Factorizations of Polynomials 1 Section IV.23. Factorizations of Polynomials over a Field Note. Our experience with classical algebra tells us that finding the zeros of a polynomial is equivalent
More informationTaylor and Maclaurin Series. Approximating functions using Polynomials.
Taylor and Maclaurin Series Approximating functions using Polynomials. Approximating f x = e x near x = 0 In order to approximate the function f x = e x near x = 0, we can use the tangent line (The Linear
More informationThe degree of the polynomial function is n. We call the term the leading term, and is called the leading coefficient. 0 =
Math 1310 A polynomial function is a function of the form = + + +...+ + where 0,,,, are real numbers and n is a whole number. The degree of the polynomial function is n. We call the term the leading term,
More informationMath 2142 Homework 5 Part 1 Solutions
Math 2142 Homework 5 Part 1 Solutions Problem 1. For the following homogeneous second order differential equations, give the general solution and the particular solution satisfying the given initial conditions.
More informationAdvanced Math Quiz Review Name: Dec Use Synthetic Division to divide the first polynomial by the second polynomial.
Advanced Math Quiz 3.1-3.2 Review Name: Dec. 2014 Use Synthetic Division to divide the first polynomial by the second polynomial. 1. 5x 3 + 6x 2 8 x + 1, x 5 1. Quotient: 2. x 5 10x 3 + 5 x 1, x + 4 2.
More information20D - Homework Assignment 4
Brian Bowers (TA for Hui Sun) MATH 0D Homework Assignment November, 03 0D - Homework Assignment First, I will give a brief overview of how to use variation of parameters. () Ensure that the differential
More informationChapter REVIEW ANSWER KEY
TEXTBOOK HELP Pg. 313 Chapter 3.2-3.4 REVIEW ANSWER KEY 1. What qualifies a function as a polynomial? Powers = non-negative integers Polynomial functions of degree 2 or higher have graphs that are smooth
More informationBob Brown Math 251 Calculus 1 Chapter 4, Section 1 Completed 1 CCBC Dundalk
Bob Brown Math 251 Calculus 1 Chapter 4, Section 1 Completed 1 Absolute (or Global) Minima and Maxima Def.: Let x = c be a number in the domain of a function f. f has an absolute (or, global ) minimum
More information2015 Math Camp Calculus Exam Solution
015 Math Camp Calculus Exam Solution Problem 1: x = x x +5 4+5 = 9 = 3 1. lim We also accepted ±3, even though it is not according to the prevailing convention 1. x x 4 x+4 =. lim 4 4+4 = 4 0 = 4 0 = We
More informationBob Brown Math 251 Calculus 1 Chapter 4, Section 4 1 CCBC Dundalk
Bob Brown Math 251 Calculus 1 Chapter 4, Section 4 1 A Function and its Second Derivative Recall page 4 of Handout 3.1 where we encountered the third degree polynomial f(x) = x 3 5x 2 4x + 20. Its derivative
More informationMath 131. The Derivative and the Tangent Line Problem Larson Section 2.1
Math 131. The Derivative and the Tangent Line Problem Larson Section.1 From precalculus, the secant line through the two points (c, f(c)) and (c +, f(c + )) is given by m sec = rise f(c + ) f(c) f(c +
More informationAnnouncements Monday, September 25
Announcements Monday, September 25 The midterm will be returned in recitation on Friday. You can pick it up from me in office hours before then. Keep tabs on your grades on Canvas. WeBWorK 1.7 is due Friday
More informationAdvanced Mathematics Unit 2 Limits and Continuity
Advanced Mathematics 3208 Unit 2 Limits and Continuity NEED TO KNOW Expanding Expanding Expand the following: A) (a + b) 2 B) (a + b) 3 C) (a + b)4 Pascals Triangle: D) (x + 2) 4 E) (2x -3) 5 Random Factoring
More informationAdvanced Mathematics Unit 2 Limits and Continuity
Advanced Mathematics 3208 Unit 2 Limits and Continuity NEED TO KNOW Expanding Expanding Expand the following: A) (a + b) 2 B) (a + b) 3 C) (a + b)4 Pascals Triangle: D) (x + 2) 4 E) (2x -3) 5 Random Factoring
More informationVANDERBILT UNIVERSITY. MATH 2300 MULTIVARIABLE CALCULUS Practice Test 1 Solutions
VANDERBILT UNIVERSITY MATH 2300 MULTIVARIABLE CALCULUS Practice Test 1 Solutions Directions. This practice test should be used as a study guide, illustrating the concepts that will be emphasized in the
More informationVector Calculus. Lecture Notes
Vector Calculus Lecture Notes Adolfo J. Rumbos c Draft date November 23, 211 2 Contents 1 Motivation for the course 5 2 Euclidean Space 7 2.1 Definition of n Dimensional Euclidean Space........... 7 2.2
More informationx =. x = x 2 x 1 x 2 x 2
3 Linear function 31 Introduction In calculus, a vector in the plane R 2 with components 2 and 3 is usually written using notation such as v = 2, 3 For our purposes it turns out to be more convenient to
More informationTaylor and Maclaurin Series. Approximating functions using Polynomials.
Taylor and Maclaurin Series Approximating functions using Polynomials. Approximating f x = e x near x = 0 In order to approximate the function f x = e x near x = 0, we can use the tangent line (The Linear
More information1. Which one of the following points is a singular point of. f(x) = (x 1) 2/3? f(x) = 3x 3 4x 2 5x + 6? (C)
Math 1120 Calculus Test 3 November 4, 1 Name In the first 10 problems, each part counts 5 points (total 50 points) and the final three problems count 20 points each Multiple choice section Circle the correct
More informationMATH 220 FINAL EXAMINATION December 13, Name ID # Section #
MATH 22 FINAL EXAMINATION December 3, 2 Name ID # Section # There are??multiple choice questions. Each problem is worth 5 points. Four possible answers are given for each problem, only one of which is
More informationHomework 8/Solutions
MTH 309-4 Linear Algebra I F11 Homework 8/Solutions Section Exercises 6.2 1,2,9,12,16,21 Section 6.2 Exercise 2. For each of the following functions, either show the function is onto by choosing an arbitrary
More informationDIFFERENTIAL EQUATIONS
DIFFERENTIAL EQUATIONS Chapter 1 Introduction and Basic Terminology Most of the phenomena studied in the sciences and engineering involve processes that change with time. For example, it is well known
More informationNotes on multivariable calculus
Notes on multivariable calculus Jonathan Wise February 2, 2010 1 Review of trigonometry Trigonometry is essentially the study of the relationship between polar coordinates and Cartesian coordinates in
More informationSpan and Linear Independence
Span and Linear Independence It is common to confuse span and linear independence, because although they are different concepts, they are related. To see their relationship, let s revisit the previous
More informationFinish section 3.6 on Determinants and connections to matrix inverses. Use last week's notes. Then if we have time on Tuesday, begin:
Math 225-4 Week 7 notes Sections 4-43 vector space concepts Tues Feb 2 Finish section 36 on Determinants and connections to matrix inverses Use last week's notes Then if we have time on Tuesday, begin
More informationMath 651 Introduction to Numerical Analysis I Fall SOLUTIONS: Homework Set 1
ath 651 Introduction to Numerical Analysis I Fall 2010 SOLUTIONS: Homework Set 1 1. Consider the polynomial f(x) = x 2 x 2. (a) Find P 1 (x), P 2 (x) and P 3 (x) for f(x) about x 0 = 0. What is the relation
More informationInvestigating Limits in MATLAB
MTH229 Investigating Limits in MATLAB Project 5 Exercises NAME: SECTION: INSTRUCTOR: Exercise 1: Use the graphical approach to find the following right limit of f(x) = x x, x > 0 lim x 0 + xx What is the
More informationMath 369 Exam #2 Practice Problem Solutions
Math 369 Exam #2 Practice Problem Solutions 2 5. Is { 2, 3, 8 } a basis for R 3? Answer: No, it is not. To show that it is not a basis, it suffices to show that this is not a linearly independent set.
More informationLinear Systems. Class 27. c 2008 Ron Buckmire. TITLE Projection Matrices and Orthogonal Diagonalization CURRENT READING Poole 5.4
Linear Systems Math Spring 8 c 8 Ron Buckmire Fowler 9 MWF 9: am - :5 am http://faculty.oxy.edu/ron/math//8/ Class 7 TITLE Projection Matrices and Orthogonal Diagonalization CURRENT READING Poole 5. Summary
More information1 The relation between a second order linear ode and a system of two rst order linear odes
Math 1280 Spring, 2010 1 The relation between a second order linear ode and a system of two rst order linear odes In Chapter 3 of the text you learn to solve some second order linear ode's, such as x 00
More informationGeorgia Department of Education Common Core Georgia Performance Standards Framework CCGPS Advanced Algebra Unit 2
Polynomials Patterns Task 1. To get an idea of what polynomial functions look like, we can graph the first through fifth degree polynomials with leading coefficients of 1. For each polynomial function,
More informationChapter 4: Interpolation and Approximation. October 28, 2005
Chapter 4: Interpolation and Approximation October 28, 2005 Outline 1 2.4 Linear Interpolation 2 4.1 Lagrange Interpolation 3 4.2 Newton Interpolation and Divided Differences 4 4.3 Interpolation Error
More informationFind all points where the function is discontinuous. 1) Find all vertical asymptotes of the given function. x(x - 1) 2) f(x) =
Math 90 Final Review Find all points where the function is discontinuous. ) Find all vertical asymptotes of the given function. x(x - ) 2) f(x) = x3 + 4x Provide an appropriate response. 3) If x 3 f(x)
More informationMath 456: Mathematical Modeling. Tuesday, March 6th, 2018
Math 456: Mathematical Modeling Tuesday, March 6th, 2018 Markov Chains: Exit distributions and the Strong Markov Property Tuesday, March 6th, 2018 Last time 1. Weighted graphs. 2. Existence of stationary
More informationTest 2 Review Math 1111 College Algebra
Test 2 Review Math 1111 College Algebra 1. Begin by graphing the standard quadratic function f(x) = x 2. Then use transformations of this graph to graph the given function. g(x) = x 2 + 2 *a. b. c. d.
More informationSolution to Set 7, Math 2568
Solution to Set 7, Math 568 S 5.: No. 18: Let Q be the set of all nonsingular matrices with the usual definition of addition and scalar multiplication. Show that Q is not a vector space. In particular,
More informationSection 14.1 Vector Functions and Space Curves
Section 14.1 Vector Functions and Space Curves Functions whose range does not consists of numbers A bulk of elementary mathematics involves the study of functions - rules that assign to a given input a
More informationExercises for Multivariable Differential Calculus XM521
This document lists all the exercises for XM521. The Type I (True/False) exercises will be given, and should be answered, online immediately following each lecture. The Type III exercises are to be done
More informationParametric Equations
Parametric Equations By: OpenStaxCollege Consider the path a moon follows as it orbits a planet, which simultaneously rotates around the sun, as seen in [link]. At any moment, the moon is located at a
More informationInfinite series, improper integrals, and Taylor series
Chapter Infinite series, improper integrals, and Taylor series. Determine which of the following sequences converge or diverge (a) {e n } (b) {2 n } (c) {ne 2n } (d) { 2 n } (e) {n } (f) {ln(n)} 2.2 Which
More informationWed Feb The vector spaces 2, 3, n. Announcements: Warm-up Exercise:
Wed Feb 2 4-42 The vector spaces 2, 3, n Announcements: Warm-up Exercise: 4-42 The vector space m and its subspaces; concepts related to "linear combinations of vectors" Geometric interpretation of vectors
More informationMath 220 Some Exam 1 Practice Problems Fall 2017
Math Some Exam Practice Problems Fall 7 Note that this is not a sample exam. This is much longer than your exam will be. However, the ideas and question types represented here (along with your homework)
More informationAIMS Exercise Set # 1
AIMS Exercise Set #. Determine the form of the single precision floating point arithmetic used in the computers at AIMS. What is the largest number that can be accurately represented? What is the smallest
More information2.3 Terminology for Systems of Linear Equations
page 133 e 2t sin 2t 44 A(t) = t 2 5 te t, a = 0, b = 1 sec 2 t 3t sin t 45 The matrix function A(t) in Problem 39, with a = 0 and b = 1 Integration of matrix functions given in the text was done with
More informationTransformations and A Universal First Order Taylor Expansion
Transformations and A Universal First Order Taylor Expansion MATH 1502 Calculus II Notes September 29, 2008 The first order Taylor approximation for f : R R at x = x 0 is given by P 1 (x) = f (x 0 )(x
More information1 Last time: multiplying vectors matrices
MATH Linear algebra (Fall 7) Lecture Last time: multiplying vectors matrices Given a matrix A = a a a n a a a n and a vector v = a m a m a mn Av = v a a + v a a v v + + Rn we define a n a n a m a m a mn
More informationENGINEERING MATH 1 Fall 2009 VECTOR SPACES
ENGINEERING MATH 1 Fall 2009 VECTOR SPACES A vector space, more specifically, a real vector space (as opposed to a complex one or some even stranger ones) is any set that is closed under an operation of
More information7.1. Calculus of inverse functions. Text Section 7.1 Exercise:
Contents 7. Inverse functions 1 7.1. Calculus of inverse functions 2 7.2. Derivatives of exponential function 4 7.3. Logarithmic function 6 7.4. Derivatives of logarithmic functions 7 7.5. Exponential
More informationECONOMICS 207 SPRING 2006 LABORATORY EXERCISE 5 KEY. 8 = 10(5x 2) = 9(3x + 8), x 50x 20 = 27x x = 92 x = 4. 8x 2 22x + 15 = 0 (2x 3)(4x 5) = 0
ECONOMICS 07 SPRING 006 LABORATORY EXERCISE 5 KEY Problem. Solve the following equations for x. a 5x 3x + 8 = 9 0 5x 3x + 8 9 8 = 0(5x ) = 9(3x + 8), x 0 3 50x 0 = 7x + 7 3x = 9 x = 4 b 8x x + 5 = 0 8x
More informationSection Properties of Rational Expressions
88 Section. - Properties of Rational Expressions Recall that a rational number is any number that can be written as the ratio of two integers where the integer in the denominator cannot be. Rational Numbers:
More informationMATH 260 LINEAR ALGEBRA EXAM III Fall 2014
MAH 60 LINEAR ALGEBRA EXAM III Fall 0 Instructions: the use of built-in functions of your calculator such as det( ) or RREF is permitted ) Consider the table and the vectors and matrices given below Fill
More informationCalculus for the Life Sciences II Assignment 6 solutions. f(x, y) = 3π 3 cos 2x + 2 sin 3y
Calculus for the Life Sciences II Assignment 6 solutions Find the tangent plane to the graph of the function at the point (0, π f(x, y = 3π 3 cos 2x + 2 sin 3y Solution: The tangent plane of f at a point
More informationSpring 2014 Math 272 Final Exam Review Sheet
Spring 2014 Math 272 Final Exam Review Sheet You will not be allowed use of a calculator or any other device other than your pencil or pen and some scratch paper. Notes are also not allowed. In kindness
More informationChapter 4E - Combinations of Functions
Fry Texas A&M University!! Math 150!! Chapter 4E!! Fall 2015! 121 Chapter 4E - Combinations of Functions 1. Let f (x) = 3 x and g(x) = 3+ x a) What is the domain of f (x)? b) What is the domain of g(x)?
More informationExercises Chapter II.
Page 64 Exercises Chapter II. 5. Let A = (1, 2) and B = ( 2, 6). Sketch vectors of the form X = c 1 A + c 2 B for various values of c 1 and c 2. Which vectors in R 2 can be written in this manner? B y
More informationEXERCISES ON DETERMINANTS, EIGENVALUES AND EIGENVECTORS. 1. Determinants
EXERCISES ON DETERMINANTS, EIGENVALUES AND EIGENVECTORS. Determinants Ex... Let A = 0 4 4 2 0 and B = 0 3 0. (a) Compute 0 0 0 0 A. (b) Compute det(2a 2 B), det(4a + B), det(2(a 3 B 2 )). 0 t Ex..2. For
More informationRational Functions. Elementary Functions. Algebra with mixed fractions. Algebra with mixed fractions
Rational Functions A rational function f (x) is a function which is the ratio of two polynomials, that is, Part 2, Polynomials Lecture 26a, Rational Functions f (x) = where and are polynomials Dr Ken W
More informationReview Questions for Test 3 Hints and Answers
eview Questions for Test 3 Hints and Answers A. Some eview Questions on Vector Fields and Operations. A. (a) The sketch is left to the reader, but the vector field appears to swirl in a clockwise direction,
More informationMath 113 HW #10 Solutions
Math HW #0 Solutions 4.5 4. Use the guidelines of this section to sketch the curve Answer: Using the quotient rule, y = x x + 9. y = (x + 9)(x) x (x) (x + 9) = 8x (x + 9). Since the denominator is always
More information17. C M 2 (C), the set of all 2 2 matrices with complex entries. 19. Is C 3 a real vector space? Explain.
250 CHAPTER 4 Vector Spaces 14. On R 2, define the operation of addition by (x 1,y 1 ) + (x 2,y 2 ) = (x 1 x 2,y 1 y 2 ). Do axioms A5 and A6 in the definition of a vector space hold? Justify your answer.
More informationMSM120 1M1 First year mathematics for civil engineers Revision notes 3
MSM0 M First year mathematics for civil engineers Revision notes Professor Robert. Wilson utumn 00 Functions Definition of a function: it is a rule which, given a value of the independent variable (often
More information1.1 Single Variable Calculus versus Multivariable Calculus Rectangular Coordinate Systems... 4
MATH2202 Notebook 1 Fall 2015/2016 prepared by Professor Jenny Baglivo Contents 1 MATH2202 Notebook 1 3 1.1 Single Variable Calculus versus Multivariable Calculus................... 3 1.2 Rectangular Coordinate
More informationLesson 59 Rolle s Theorem and the Mean Value Theorem
Lesson 59 Rolle s Theorem and the Mean Value Theorem HL Math - Calculus After this lesson, you should be able to: Understand and use Rolle s Theorem Understand and use the Mean Value Theorem 1 Rolle s
More informationv(t) v(t) Assignment & Notes 5.2: Intro to Integrals Due Date: Friday, 1/10
Assignment & Notes 5.2: Intro to Integrals 1. The velocity function (in miles and hours) for Ms. Hardtke s Christmas drive to see her family is shown at the right. Find the total distance Ms. H travelled
More informationMath 113 Winter 2005 Departmental Final Exam
Name Student Number Section Number Instructor Math Winter 2005 Departmental Final Exam Instructions: The time limit is hours. Problem consists of short answer questions. Problems 2 through are multiple
More information