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School: UCSB
Homework 1 Solutions 1. (#1.1.2 in Strauss) Which of the following operators are linear? (a) Lu = ux + xuy (b) Lu = ux + uuy (c) Lu = ux + u2 y (d) Lu = ux + uy + 1 (e) Lu = 1 + x2 (cos y )ux + uyxy [arctan(x/y )]u Solution: (a) Linear. (b) Nonlinear the
School: UCSB
Course: Calc With Appli 2
MATH 3B EXAM I PRACTICE January 13, 2011 JEFFREY STOPPLE In these notes you will come up with your own practice exam questions. This will better help internalize the material. You should solve your own exam, or if you have a study partner you should switc
School: UCSB
Course: 117
Homework 2 Hctor Guillermo Cullar R e e os February 2, 2006 12.12 Let D be a nonempty set and suppose that f : D R and g : D R. Dene the function f + g : D R by (f + g)(x) = f (x) + g(x). (a) If f (D) and g(D) are bounded above, then prove that (f
School: UCSB
Homework 5 Math 104A, Fall 2010 Due on Tuesday, November 9th, 2010 1. Given xi , i = 0, 1, . . . , n, consider the Lagrange polynomials Ln,j for j = 0, 1, . . . , n. Prove that n Ln,j (x) = 1 for all x R. j =0 2. The following data is taken from a polynom
School: UCSB
Course: Vector Calculus 2
Math 5C: Exam #2 Solutions Date: July 16th , 2010 Score: out of 60 1. (10) Match each Maclaurin series to the function from the following list it represents by lling in the blank space below the series. (Note: All listed function are C at x = 0 under the
School: UCSB
Course: Probability Theory And Stochastic Processes
PStat 213A: Probability Theory and Stochastic Processes Simon RubinsteinSalzedo Fall 2005 0.1 Introduction These notes are based on a graduate course on probability theory and stochastic processes I took from Professor Raya Feldman in the Fall of 2005. Th
School: UCSB
School: UCSB
School: UCSB
School: UCSB
School: UCSB
School: UCSB
Math 34A Lecture 17 Copyright Daryl Cooper D.A.R.Y.L. Please do NOT come on stage November 9, 2012 Homework 7.13.43 Some biologists at UCSB have carefully recorded the number of elephant seal births in the Channel Islands from aerial photographs since the
School: UCSB
Course: Math 4A
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School: UCSB
Course: Math 4A
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School: UCSB
Course: Math 4A
Wolves & Rabbits Eigenvectors and Eigenvalues Finding Eigenvalues Finding Eigenvectors Eigenspaces Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 20, Feb. 28 2014 Based on the 2013 Millett and Scharlemann Lectures 1/21 Wolves & Rabbits
School: UCSB
Course: Math 4A
Last Time: Dot Product Distance Orthogonality Orthogonal sets Weights(coordinates) from orthogonal basis Orthonormal sets Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 23, March 7 2014 Based on the 2013 Millett and Scharlemann Lectures
School: UCSB
Course: Math 4A
Last Time: Eigenvalues and Eigenvectors Diagonalization Powers To the Matrix Diagonalization Again Triangular example Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 21, March 3 2014 Based on the 2013 Millett and Scharlemann Lectures 1/2
School: UCSB
Course: Calc With Appli 2
MATH 3B EXAM I PRACTICE January 13, 2011 JEFFREY STOPPLE In these notes you will come up with your own practice exam questions. This will better help internalize the material. You should solve your own exam, or if you have a study partner you should switc
School: UCSB
Course: Vector Calculus 2
Math 5C: Exam #2 Solutions Date: July 16th , 2010 Score: out of 60 1. (10) Match each Maclaurin series to the function from the following list it represents by lling in the blank space below the series. (Note: All listed function are C at x = 0 under the
School: UCSB
Course: Calculus
Being the final examination for Math 3B NO notes or calculators. READ all questions carefully. Make sure your answers are clearly marked and it is clear what work is relevant and should be graded. Each problem is worth 20 points. Note there is a blank pag
School: UCSB
Introduction to Numerical Analysis Math 104A, Winter 2009 Instructor: Carlos J. Garc a-Cervera March 17th, 2009 Answer the following 8 questions. Show all your work for full credit. You must include your computer programs. Follow the guidelines for presen
School: UCSB
Introduction to Numerical Analysis Math 104A, Winter 2009 Instructor: Carlos J. Garc a-Cervera December 8th, 2010 Answer the following 7 questions. Show all your work for full credit. You must include your computer programs. Follow the guidelines for pres
School: UCSB
M A T H 3 4A Sp r i n g 1 4 Yu a n q i A pr i . Yo u r D ay P erm 2 . 4 35 \ / D Co p y i n NO On ly H o r t isc u s sio n ex p l ai n y ou r g so m eo n e 8 an s w e r a n d c h ec k y o u r s o l u t i o n s else s t e s t , o r an d d e l i b e r a t
School: UCSB
Homework 1 Solutions 1. (#1.1.2 in Strauss) Which of the following operators are linear? (a) Lu = ux + xuy (b) Lu = ux + uuy (c) Lu = ux + u2 y (d) Lu = ux + uy + 1 (e) Lu = 1 + x2 (cos y )ux + uyxy [arctan(x/y )]u Solution: (a) Linear. (b) Nonlinear the
School: UCSB
Course: 117
Homework 2 Hctor Guillermo Cullar R e e os February 2, 2006 12.12 Let D be a nonempty set and suppose that f : D R and g : D R. Dene the function f + g : D R by (f + g)(x) = f (x) + g(x). (a) If f (D) and g(D) are bounded above, then prove that (f
School: UCSB
Homework 5 Math 104A, Fall 2010 Due on Tuesday, November 9th, 2010 1. Given xi , i = 0, 1, . . . , n, consider the Lagrange polynomials Ln,j for j = 0, 1, . . . , n. Prove that n Ln,j (x) = 1 for all x R. j =0 2. The following data is taken from a polynom
School: UCSB
Course: Graph Theory
HOMEWORK 3 SOLUTIONS (1) Show that for each n N the complete graph Kn is a contraction of Kn,n . Solution: We describe the process for several small values of n. In this way, we can discern the inductive step. Clearly, K1 , which is just one vertex, is a
School: UCSB
Robin Meister Homework Set HW11 due 10/25/2013 at 05:00am PDT MATH 34A Fall 2012 !Correct Answers: Cooper 2.2.11 (5)/(70-5-55) (5)/(v-55-5) 1. (1 pt) To do this question you need to know that speed is distance travelled divided by time taken. Three obje
School: UCSB
School: UCSB
School: UCSB
School: UCSB
School: UCSB
School: UCSB
School: UCSB
Course: Math 4A
Math 4A Syllabus Winter 2014 Lecture: MWF 10:00 10:50am, MUSIC LLCH Text: Linear Algebra with applications by David C. Lay, Addison-Wesley, 4th Edition. Material to be covered: Chapters 1-6 in the text book. iClicker: You should purchase an iClicker and b
School: UCSB
Math 108a Professor: Padraic Bartlett Syllabus for Math 108a Weeks 1-10 UCSB 2013 Basic Course Information Professor: Padraic Bartlett. Class time/location: MWF 9-9:50, Phelps 3505. Oce hours/location: TTh 2-3pm, South Hall 6516. Additionally, I am tea
School: UCSB
Analysis with an introduction to proof - Math 117 Spring2009 Monday, Wednesday, & Friday, 12:00-12:50pm, South Hall 6635 Instructor: Carlos J. Garc a-Cervera. Oce: South Hall, 6707. Phone: 893-3681. E-mail: cgarcia@math.ucsb.edu URL: http:/www.math.ucsb.e
School: UCSB
Analysis with an introduction to proof - Math 117 Spring2009 Monday, Wednesday, & Friday, 12:00-12:50pm, South Hall 6635 Instructor: Carlos J. Garc a-Cervera. Oce: South Hall, 6707. Phone: 893-3681. E-mail: cgarcia@math.ucsb.edu URL: http:/www.math.ucsb.e
School: UCSB
Intro. to Numerical Analysis - Math 104B Winter 2011 Tuesday & Thursday, 8:00-9:15am, South Hall 6635 Instructor: Carlos J. Garc a-Cervera. Oce: South Hall, 6707. Phone: 893-3681. E-mail: cgarcia@math.ucsb.edu URL: http:/www.math.ucsb.edu/~cgarcia/Courses
School: UCSB
Math 104A, Fall 2010 Intro. to Numerical Analysis Tuesday & Thursday, 9:30-10:45am, 387 101 Instructor: Carlos J. Garc a-Cervera. Oce: South Hall, 6707. Phone: (805) 893-3681. E-mail: cgarcia@math.ucsb.edu URL: http:/www.math.ucsb.edu/~cgarcia/Courses/Mat
School: UCSB
Homework 1 Solutions 1. (#1.1.2 in Strauss) Which of the following operators are linear? (a) Lu = ux + xuy (b) Lu = ux + uuy (c) Lu = ux + u2 y (d) Lu = ux + uy + 1 (e) Lu = 1 + x2 (cos y )ux + uyxy [arctan(x/y )]u Solution: (a) Linear. (b) Nonlinear the
School: UCSB
Course: Calc With Appli 2
MATH 3B EXAM I PRACTICE January 13, 2011 JEFFREY STOPPLE In these notes you will come up with your own practice exam questions. This will better help internalize the material. You should solve your own exam, or if you have a study partner you should switc
School: UCSB
Course: 117
Homework 2 Hctor Guillermo Cullar R e e os February 2, 2006 12.12 Let D be a nonempty set and suppose that f : D R and g : D R. Dene the function f + g : D R by (f + g)(x) = f (x) + g(x). (a) If f (D) and g(D) are bounded above, then prove that (f
School: UCSB
Homework 5 Math 104A, Fall 2010 Due on Tuesday, November 9th, 2010 1. Given xi , i = 0, 1, . . . , n, consider the Lagrange polynomials Ln,j for j = 0, 1, . . . , n. Prove that n Ln,j (x) = 1 for all x R. j =0 2. The following data is taken from a polynom
School: UCSB
Course: Vector Calculus 2
Math 5C: Exam #2 Solutions Date: July 16th , 2010 Score: out of 60 1. (10) Match each Maclaurin series to the function from the following list it represents by lling in the blank space below the series. (Note: All listed function are C at x = 0 under the
School: UCSB
Course: Probability Theory And Stochastic Processes
PStat 213A: Probability Theory and Stochastic Processes Simon RubinsteinSalzedo Fall 2005 0.1 Introduction These notes are based on a graduate course on probability theory and stochastic processes I took from Professor Raya Feldman in the Fall of 2005. Th
School: UCSB
Course: Calculus
Being the final examination for Math 3B NO notes or calculators. READ all questions carefully. Make sure your answers are clearly marked and it is clear what work is relevant and should be graded. Each problem is worth 20 points. Note there is a blank pag
School: UCSB
Course: Differential Equations
-7 -3 -1 1 -2 26.(1 pt) 0 -2 -1 -2 -28 -12 -6 3 . 32.(1 pt) Let M = -2 3 Compute the rank of the above matrix -2 2 1 Find c1 , c2 , and c3 such that M 3 + c1 M 2 + c2 M + c3 I3 = 0, where 7 4 7 I3 is the identity 3 3 matrix. 27.(1 pt) 7 4 3 , c1 = 21 12
School: UCSB
Course: Graph Theory
HOMEWORK 3 SOLUTIONS (1) Show that for each n N the complete graph Kn is a contraction of Kn,n . Solution: We describe the process for several small values of n. In this way, we can discern the inductive step. Clearly, K1 , which is just one vertex, is a
School: UCSB
Introduction to Numerical Analysis Math 104A, Winter 2009 Instructor: Carlos J. Garc a-Cervera March 17th, 2009 Answer the following 8 questions. Show all your work for full credit. You must include your computer programs. Follow the guidelines for presen
School: UCSB
Introduction to Numerical Analysis Math 104A, Winter 2009 Instructor: Carlos J. Garc a-Cervera December 8th, 2010 Answer the following 7 questions. Show all your work for full credit. You must include your computer programs. Follow the guidelines for pres
School: UCSB
Robin Meister Homework Set HW11 due 10/25/2013 at 05:00am PDT MATH 34A Fall 2012 !Correct Answers: Cooper 2.2.11 (5)/(70-5-55) (5)/(v-55-5) 1. (1 pt) To do this question you need to know that speed is distance travelled divided by time taken. Three obje
School: UCSB
Robin Meister Homework Set HW13 due 10/30/2013 at 06:00am PDT MATH 34A Fall 2012 a)7100 1. (1 pt) Cooper 7.13.6 Use logs and antilogs to perform the following calculations. DO NOT JUST MULTIPLY THE WAY YOU LEARNED IN 3rd GRADE OR USE A CALCULATOR! a)335 4
School: UCSB
Robin Meister Homework Set HW18 due 11/13/2013 at 05:00am PST MATH 34A Fall 2012 !Correct Answers: 2*50-52 Cooper 3.2.15 1. (1 pt) (a) A cube has surface area 294m2 . What is the volume of the cube? Volume of Cube= m3 Cooper 3.2.32 5. (1 pt) The number N
School: UCSB
Math 34A Lecture 17 Copyright Daryl Cooper D.A.R.Y.L. Please do NOT come on stage November 9, 2012 Homework 7.13.43 Some biologists at UCSB have carefully recorded the number of elephant seal births in the Channel Islands from aerial photographs since the
School: UCSB
Robin Meister Homework Set HW8 due 10/18/2013 at 05:00am PDT MATH 34A Fall 2012 Cooper 1.5.32 1. (1 pt) If f is the focal length of a lens and u is the distance of an object from the lens and v is the distance of the image from the lens then 1 1 1 + =. u
School: UCSB
Robin Meister Homework Set HW7 due 10/16/2013 at 05:00am PDT MATH 34A Fall 2012 (a) How far apart are they after 1 hour? 1. (1 pt) Cooper 1.5.9 (b) How far apart are they after t hours? (c) When are they 4160 miles apart? a) miles Solve (x 10)(x 20)(x 30)
School: UCSB
October 15, 2011 13:47 ast Sheet number 3 Page number xx cyan magenta yellow black October 10, 2011 15:10 frs Sheet number 3 Page number iii cyan magenta yellow black 10 th EDITION David Henderson/Getty Images CALCULUS EARLY TRANSCENDENTALS HOWARD ANTON I
School: UCSB
Course: Differential Equations
Sample WeBWorK problems. 1.(1 pt) Which of the following are vectors in R2 ? A. (1,0,0) B. (1,0) C. x2 D. x E. 1 F. (0,1) G. None of the above Which of the following is the zero vector in R2 ? A. 0 B. (0,0,0) C. (0,0) D. None of the above Which pairs belo
School: UCSB
Course: Math 4A
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School: UCSB
Course: Math 4A
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Course: Math 4A
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Course: Math 4A
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School: UCSB
Course: Math 4A
Wolves & Rabbits Eigenvectors and Eigenvalues Finding Eigenvalues Finding Eigenvectors Eigenspaces Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 20, Feb. 28 2014 Based on the 2013 Millett and Scharlemann Lectures 1/21 Wolves & Rabbits
School: UCSB
Course: Math 4A
Last Time: Dot Product Distance Orthogonality Orthogonal sets Weights(coordinates) from orthogonal basis Orthonormal sets Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 23, March 7 2014 Based on the 2013 Millett and Scharlemann Lectures
School: UCSB
Course: Math 4A
Math 4A Syllabus Winter 2014 Lecture: MWF 10:00 10:50am, MUSIC LLCH Text: Linear Algebra with applications by David C. Lay, Addison-Wesley, 4th Edition. Material to be covered: Chapters 1-6 in the text book. iClicker: You should purchase an iClicker and b
School: UCSB
Course: Math 4A
Last Time: Eigenvalues and Eigenvectors Diagonalization Powers To the Matrix Diagonalization Again Triangular example Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 21, March 3 2014 Based on the 2013 Millett and Scharlemann Lectures 1/2
School: UCSB
Course: Math 4A
Last Time: Distance and Orthogonality Least Square Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 24, March 10 2014 Based on the 2013 Millett and Scharlemann Lectures 1/12 Last Time: Distance and Orthogonality Least Square Last Time: Di
School: UCSB
Course: Math 4A
Last Time: Diagonalization Thinking Complex Repeat! Dot product dened Dot product properties Vector length Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 22, March 5 2014 Based on the 2013 Millett and Scharlemann Lectures 1/14 Last Time
School: UCSB
Course: Math 4A
Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 19, Feb. 24 2014 Based on the 2013 Millett and Scharlemann Lectures 1/1 Last Time: Coordinates w.r.t. a Basis Suppose v1 , . . . vk is a basis for V. For any v V there is exactly one set c1
School: UCSB
Course: Math 4A
Last Time: Column Space, Image, Finding Basis Basis Again Coordinates Changing basis and coordinates in Rn Midterm Discuss Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 18, Feb. 21 2014 Based on the 2013 Millett and Scharlemann Lecture
School: UCSB
Course: Math 4A
Last Time: Null, Kernel, Span, Basis Column space of a matrix Image of linear transformation Back to Basis Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 17, Feb. 19 2014 Based on the 2013 Millett and Scharlemann Lectures 1/18 Last Time
School: UCSB
Course: Math 4A
Last Time: Vector Spaces And Subspaces Null-space of a matrix Kernel of linear transformation Linear Independence And Basis C Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 16, Feb. 14 2014 Based on the 2013 Millett and Scharlemann Lect
School: UCSB
Course: Math 4A
Last Time: Determinant via EROs More About Determinant The Geometry Going Abstract Dene & describe vector spaces Sub Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 14, Feb. 10 2014 Based on the 2013 Millett and Scharlemann Lectures 1/24
School: UCSB
Course: Math 4A
Last Time: Linear Independence Solutions of Homogeneous Systems Solutions of Non-homogeneous Systems Linear Transformatio Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 8, Jan. 24 2014 Based on the 2013 Millett and Scharlemann Lectures
School: UCSB
Course: Math 4A
Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 9, Jan. 27 2014 Based on the 2013 Millett and Scharlemann Lectures 1/1 Last Time: Linear Independence, Solution Sets In general, to see if a set of vectors cfw_a1 , . . . , an is linearly
School: UCSB
Course: Math 4A
Last Time: Vector Spaces Subspaces Examples and non-examples in Rn Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 15, Feb. 12 2014 Based on the 2013 Millett and Scharlemann Lectures 1/14 Last Time: Vector Spaces Subspaces Examples and n
School: UCSB
Course: Math 4A
Last Time: Invertible Matrix, Determinant Triangular And Almost Triangular Determinant And EROs Determinants via EROs Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 13, Feb. 7 2014 Based on the 2013 Millett and Scharlemann Lectures 1/22
School: UCSB
Course: Math 4A
Last Time: How to Invert Matrices The Core Theorem Tying Together Equivalent triple All tied together Determinants Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 12, Feb. 5 2014 Based on the 2013 Millett and Scharlemann Lectures 1/21 La
School: UCSB
Course: Math 4A
Last Time: Matrix of Linear Transformation One-to-One, Onto Matrix Algebra Invertible Matrices Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 10, Jan. 31 2014 Based on the 2013 Millett and Scharlemann Lectures 1/18 Last Time: Matrix of
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Course: Math 4A
Last Time: Matrix Inverse How to See Inverse The Core Theorem Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 11, Feb. 2014 Based on the 2013 Millett and Scharlemann Lectures 1/11 Last Time: Matrix Inverse How to See Inverse The Core The
School: UCSB
Course: Math 4A
Last Time: Span and Matrix-vector Multiplication Matrix Multiplication Back to SpanLinear Independence How to Determine (I Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 7, Jan. 22 2014 Based on the 2013 Millett and Scharlemann Lectures
School: UCSB
Course: Math 4A
Last Time: Echelon is Cool Vectors Picturing Vectors Properties of vectors Linear combination Span Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 5, Jan. 15 2014 Based on the 2013 Millett and Scharlemann Lectures 1/26 Last Time: Echelon
School: UCSB
Course: Math 4A
Last Time: Linear Combination And Span Matrix-vector multiplication Matrix equation Spanning the Whole Space Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 6, Jan. 17 2014 Based on the 2013 Millett and Scharlemann Lectures 1/20 Last Tim
School: UCSB
Course: Math 4A
Last Time: Solving System of Linear Equations Getting to Echelon Echelon examples (Reduced) echelon is cool Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 3, Jan. 10 2014 Based on the 2013 Millett and Scharlemann Lectures 1/16 Last Time
School: UCSB
Course: Math 4A
Last Time: Getting to Echelon (Reduced) echelon is cool Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 4, Jan. 13 2014 Based on the 2013 Millett and Scharlemann Lectures 1/7 Last Time: Getting to Echelon (Reduced) echelon is cool Gettin
School: UCSB
Course: Math 4A
Last Time: System of Linear Equations Solving a linear system Some examples Getting to echelon Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 2, Jan. 8 2014 Based on the 2013 Millett and Scharlemann Lectures 1/9 Last Time: System of Lin
School: UCSB
Course: Math 4A
Game Plan Whats Linear Algebra About? Linear Equations Systems of linear equations Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 1, Jan. 6 2014 Based on the 2013 Millett and Scharlemann Lectures 1/14 Game Plan Whats Linear Algebra Abou
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M A T H 3 4A Sp r i n g 1 4 Yu a n q i A pr i . Yo u r D ay P erm 2 . 4 35 \ / D Co p y i n NO On ly H o r t isc u s sio n ex p l ai n y ou r g so m eo n e 8 an s w e r a n d c h ec k y o u r s o l u t i o n s else s t e s t , o r an d d e l i b e r a t
School: UCSB
School: UCSB
School: UCSB
School: UCSB
School: UCSB
M A T H 3 4A Sp r i n g 1 4 Yu a n q i A pr i . Yo u r D ay P erm 2 . 4 35 \ / D C lea r l y Co p y i n NO On ly H o r t isc u s sio n ex p l ai n y ou r g so m eo n e 8 an s w e r a n d c h ec k y o u r s o l u t i o n s s t e s t , o r an d d e l i b e
School: UCSB
School: UCSB
School: UCSB
School: UCSB
School: UCSB
School: UCSB
Course: Probability Theory And Stochastic Processes
PStat 213A: Probability Theory and Stochastic Processes Simon RubinsteinSalzedo Fall 2005 0.1 Introduction These notes are based on a graduate course on probability theory and stochastic processes I took from Professor Raya Feldman in the Fall of 2005. Th
School: UCSB
School: UCSB
School: UCSB
School: UCSB
School: UCSB
School: UCSB
School: UCSB
School: UCSB
School: UCSB
School: UCSB
School: UCSB
School: UCSB
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School: UCSB
School: UCSB
School: UCSB
School: UCSB
School: UCSB
School: UCSB
School: UCSB
School: UCSB
School: UCSB
School: UCSB
School: UCSB
School: UCSB
School: UCSB
School: UCSB
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School: UCSB
School: UCSB
Course: Modern Algebra
Math 220A: Modern Algebra Simon RubinsteinSalzedo January 31, 2004 0.1 Introduction Professor: Adebisi Agboola. Oce Hours: Tuesday: 11:15-12:30, Wednesday: 11:15-12:30, Thursday: 11:15-12:30. Textbooks: Abstract Algebra by Dummit and Foote and Algebra by
School: UCSB
Course: Differential Geometry
Dierential Geometry Simon Rubinstein-Salzedo December 13, 2005 0.1 Introduction These notes are based on an undergraduate course in dierential geometry I took from Dr. Michael Crandall in the winter and spring of 2004. The primary textbook for that class
School: UCSB
Course: Modern Algebra
Math 220B: Modern Algebra Simon RubinsteinSalzedo March 11, 2004 0.1 Introduction Professor: Adebisi Agboola Oce Hours: Tuesday: 11:15-12:30, Wednesday: 11:15-12:30, Thursday: 11:15-12:30. 1 Chapter 1 Rings Denition 1.1 A ring (R, +, ) is an abelian group
School: UCSB
Course: Calculus On Manifolds
CCS Math 120: Calculus on Manifolds Simon RubinsteinSalzedo Spring 2004 0.1 Introduction These notes are based on a course on calculus on manifolds I took from Professor Martin Scharlemann in the Spring of 2004. The course was designed for rst-year CCS ma
School: UCSB
Course: Modern Algebra
Math 220C: Modern Algebra Simon RubinsteinSalzedo June 16, 2004 0.1 Introduction Professor: Adebisi Agboola Oce Hours: Tuesday: 11:15-12:30, Wednesday: 11:15-12:30, Thursday: 11:15-12:30. 1 Chapter 1 Preliminaries Question How can we tell whether a polyno
School: UCSB
Course: Modern Algebra
Group Theory Simon Rubinstein-Salzedo November 19, 2005 0.1 Introduction These notes are based on a graduate course in group theory I took from Dr. Ken Goodearl in the fall of 2004. The primary textbook for that class was Algebra by Larry Grove. Some mate
School: UCSB
Course: Game Theory
Econ 210B: Game Theory Simon RubinsteinSalzedo Fall 2005 0.1 Introduction These notes are based on a graduate course on game theory I took from Professor Rod Garratt in the Fall of 2005. The primary textbooks were Game Theory by Drew Fudenberg and Jean Ti
School: UCSB
Course: Differential Geometry
Math 240A: Dierentiable Manifolds and Riemannian Geometry Simon RubinsteinSalzedo Fall 2005 0.1 Introduction These notes are based on a graduate course on dierentiable manifolds and Riemannian geometry I took from Professor Doug Moore in the Fall of 2005.
School: UCSB
Course: Noncumulative Noetherian Rings
Math 260Q: Noetherian Rings Simon Rubinstein-Salzedo Fall 2005 0.1 Introduction These notes are based on a graduate course on noetherian rings I took from Professor Ken Goodearl in the Fall of 2005. The textbook was An Introduction to Noncommutative Noeth
School: UCSB
Course: Algebraic Number Theory
Math 225AB: Algebraic Number Theory Simon RubinsteinSalzedo Winter and Spring 2006 0.1 Introduction Professor: Adebisi Agboola. Oce Hours: Tuesday: 11:15-12:30, Thursday: 11:15-12:30 (225A), Wednesday 10:0012:00 (225B). Textbooks: Algebraic Number Theory
School: UCSB
Course: Differential Geometry
Math 240B: Dierentiable Manifolds and Riemannian Geometry Simon RubinsteinSalzedo Winter 2006 0.1 Introduction These notes are based on a graduate course on dierentiable manifolds and Riemannian geometry I took from Professor Doug Moore in the Winter of 2
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Math 240C: Dierentiable Manifolds and Riemannian Geometry Simon RubinsteinSalzedo Spring 2006 0.1 Introduction These notes are based on a graduate course on dierentiable manifolds and Riemannian geometry I took from Professor Doug Moore in the Spring of 2
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Course: Algebraic Number Theory
Math 225AB: Elliptic Curves Simon Rubinstein-Salzedo Winter and Spring 2007 0.1 Introduction These notes are based on a graduate course on elliptic curves I took from Professor Adebisi Agboola in the Winter and Spring of 2007. The textbooks were The Arith
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Course: Vector Calculus I
Math 5B, Final Review Topics and Problems Fall 2006 Here is a brief list of the key topics and important formulas we have covered since the last midterm. You should know how to do each thing listed and/or know the denitions of the key concepts. The number
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Math 34A Lecture 17 Copyright Daryl Cooper D.A.R.Y.L. Please do NOT come on stage November 9, 2012 Homework 7.13.43 Some biologists at UCSB have carefully recorded the number of elephant seal births in the Channel Islands from aerial photographs since the
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Wolves & Rabbits Eigenvectors and Eigenvalues Finding Eigenvalues Finding Eigenvectors Eigenspaces Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 20, Feb. 28 2014 Based on the 2013 Millett and Scharlemann Lectures 1/21 Wolves & Rabbits
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Last Time: Dot Product Distance Orthogonality Orthogonal sets Weights(coordinates) from orthogonal basis Orthonormal sets Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 23, March 7 2014 Based on the 2013 Millett and Scharlemann Lectures
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Last Time: Eigenvalues and Eigenvectors Diagonalization Powers To the Matrix Diagonalization Again Triangular example Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 21, March 3 2014 Based on the 2013 Millett and Scharlemann Lectures 1/2
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Last Time: Distance and Orthogonality Least Square Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 24, March 10 2014 Based on the 2013 Millett and Scharlemann Lectures 1/12 Last Time: Distance and Orthogonality Least Square Last Time: Di
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Last Time: Diagonalization Thinking Complex Repeat! Dot product dened Dot product properties Vector length Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 22, March 5 2014 Based on the 2013 Millett and Scharlemann Lectures 1/14 Last Time
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Course: Math 4A
Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 19, Feb. 24 2014 Based on the 2013 Millett and Scharlemann Lectures 1/1 Last Time: Coordinates w.r.t. a Basis Suppose v1 , . . . vk is a basis for V. For any v V there is exactly one set c1
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Last Time: Column Space, Image, Finding Basis Basis Again Coordinates Changing basis and coordinates in Rn Midterm Discuss Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 18, Feb. 21 2014 Based on the 2013 Millett and Scharlemann Lecture
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Last Time: Null, Kernel, Span, Basis Column space of a matrix Image of linear transformation Back to Basis Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 17, Feb. 19 2014 Based on the 2013 Millett and Scharlemann Lectures 1/18 Last Time
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Last Time: Vector Spaces And Subspaces Null-space of a matrix Kernel of linear transformation Linear Independence And Basis C Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 16, Feb. 14 2014 Based on the 2013 Millett and Scharlemann Lect
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Last Time: Determinant via EROs More About Determinant The Geometry Going Abstract Dene & describe vector spaces Sub Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 14, Feb. 10 2014 Based on the 2013 Millett and Scharlemann Lectures 1/24
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Last Time: Linear Independence Solutions of Homogeneous Systems Solutions of Non-homogeneous Systems Linear Transformatio Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 8, Jan. 24 2014 Based on the 2013 Millett and Scharlemann Lectures
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Course: Math 4A
Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 9, Jan. 27 2014 Based on the 2013 Millett and Scharlemann Lectures 1/1 Last Time: Linear Independence, Solution Sets In general, to see if a set of vectors cfw_a1 , . . . , an is linearly
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Course: Math 4A
Last Time: Vector Spaces Subspaces Examples and non-examples in Rn Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 15, Feb. 12 2014 Based on the 2013 Millett and Scharlemann Lectures 1/14 Last Time: Vector Spaces Subspaces Examples and n
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Last Time: Invertible Matrix, Determinant Triangular And Almost Triangular Determinant And EROs Determinants via EROs Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 13, Feb. 7 2014 Based on the 2013 Millett and Scharlemann Lectures 1/22
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Course: Math 4A
Last Time: How to Invert Matrices The Core Theorem Tying Together Equivalent triple All tied together Determinants Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 12, Feb. 5 2014 Based on the 2013 Millett and Scharlemann Lectures 1/21 La
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Course: Math 4A
Last Time: Matrix of Linear Transformation One-to-One, Onto Matrix Algebra Invertible Matrices Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 10, Jan. 31 2014 Based on the 2013 Millett and Scharlemann Lectures 1/18 Last Time: Matrix of
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Course: Math 4A
Last Time: Matrix Inverse How to See Inverse The Core Theorem Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 11, Feb. 2014 Based on the 2013 Millett and Scharlemann Lectures 1/11 Last Time: Matrix Inverse How to See Inverse The Core The
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Course: Math 4A
Last Time: Span and Matrix-vector Multiplication Matrix Multiplication Back to SpanLinear Independence How to Determine (I Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 7, Jan. 22 2014 Based on the 2013 Millett and Scharlemann Lectures
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Course: Math 4A
Last Time: Echelon is Cool Vectors Picturing Vectors Properties of vectors Linear combination Span Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 5, Jan. 15 2014 Based on the 2013 Millett and Scharlemann Lectures 1/26 Last Time: Echelon
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Course: Math 4A
Last Time: Linear Combination And Span Matrix-vector multiplication Matrix equation Spanning the Whole Space Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 6, Jan. 17 2014 Based on the 2013 Millett and Scharlemann Lectures 1/20 Last Tim
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Course: Math 4A
Last Time: Solving System of Linear Equations Getting to Echelon Echelon examples (Reduced) echelon is cool Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 3, Jan. 10 2014 Based on the 2013 Millett and Scharlemann Lectures 1/16 Last Time
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Course: Math 4A
Last Time: Getting to Echelon (Reduced) echelon is cool Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 4, Jan. 13 2014 Based on the 2013 Millett and Scharlemann Lectures 1/7 Last Time: Getting to Echelon (Reduced) echelon is cool Gettin
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Course: Math 4A
Last Time: System of Linear Equations Solving a linear system Some examples Getting to echelon Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 2, Jan. 8 2014 Based on the 2013 Millett and Scharlemann Lectures 1/9 Last Time: System of Lin
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Course: Math 4A
Game Plan Whats Linear Algebra About? Linear Equations Systems of linear equations Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 1, Jan. 6 2014 Based on the 2013 Millett and Scharlemann Lectures 1/14 Game Plan Whats Linear Algebra Abou
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Course: Calc With Appli 2
MATH 3B EXAM I PRACTICE January 13, 2011 JEFFREY STOPPLE In these notes you will come up with your own practice exam questions. This will better help internalize the material. You should solve your own exam, or if you have a study partner you should switc
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Course: Vector Calculus 2
Math 5C: Exam #2 Solutions Date: July 16th , 2010 Score: out of 60 1. (10) Match each Maclaurin series to the function from the following list it represents by lling in the blank space below the series. (Note: All listed function are C at x = 0 under the
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Course: Calculus
Being the final examination for Math 3B NO notes or calculators. READ all questions carefully. Make sure your answers are clearly marked and it is clear what work is relevant and should be graded. Each problem is worth 20 points. Note there is a blank pag
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Introduction to Numerical Analysis Math 104A, Winter 2009 Instructor: Carlos J. Garc a-Cervera March 17th, 2009 Answer the following 8 questions. Show all your work for full credit. You must include your computer programs. Follow the guidelines for presen
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Introduction to Numerical Analysis Math 104A, Winter 2009 Instructor: Carlos J. Garc a-Cervera December 8th, 2010 Answer the following 7 questions. Show all your work for full credit. You must include your computer programs. Follow the guidelines for pres
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M A T H 3 4A Sp r i n g 1 4 Yu a n q i A pr i . Yo u r D ay P erm 2 . 4 35 \ / D Co p y i n NO On ly H o r t isc u s sio n ex p l ai n y ou r g so m eo n e 8 an s w e r a n d c h ec k y o u r s o l u t i o n s else s t e s t , o r an d d e l i b e r a t
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M A T H 3 4A Sp r i n g 1 4 Yu a n q i A pr i . Yo u r D ay P erm 2 . 4 35 \ / D C lea r l y Co p y i n NO On ly H o r t isc u s sio n ex p l ai n y ou r g so m eo n e 8 an s w e r a n d c h ec k y o u r s o l u t i o n s s t e s t , o r an d d e l i b e
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Course: Math 4A
Activity 6 Try to do each of the following problems with just your note card or no notes, as if it was the midterm. Question 1. Consider the following system of equations: xyz = 1 2x + 4y + z = a x 4y + bz = 3 1. Find all the values of a and b for which t
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Course: Math 4A
Mathematics 6A Winter 2013: Review for Midterm 2 February 20, 2013 Professor J. Douglas Moore Part I. Multiple Choice. There will be several multiple choice questions in which you need to circle the correct answer. These will be similar to i-Clicker quest
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Course: Math 4A
Mathematics 6A Winter 2013: Review for Final Exam March 15, 2013 Professor J. Douglas Moore Part I. Multiple Choice. There will be several multiple choice questions in which you need to circle the correct answer. Some examples can be found as i-Clicker qu
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MATH. 117 ANSWER KEY FOR SECOND MIDTERM FEB. 2014 1. [18 pts] Fill in complete and accurate denitions (not just notation) for each term. (Do not give the conditions from some theorem either, only the basic denition, as stated when the term was rst introdu
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MATH. 117 ANSWER KEY FOR FIRST MIDTERM FEB. 2014 1. [18 pts] Fill in complete and accurate denitions (not just notation) for each term. (Do not give the conditions from some theorem either, only the basic denition, as stated when the term was rst introduc
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Mathematics*108A*Midterm* * February*13,*2014* * Place*all*answers*in*your*BlueBook.*In*order*to*receive*complete*credit,*provide* complete*clearly*and*correctly*written*answers.*T/F*and*short*answers*must*be* include*short*explanations.* Section*I*T/R*an
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Math 108A Winter Quarter 2014 Practice Problems for Final No notes or calculators are permitted on this exam. You must show all work, provide complete proofs, and provide short by complete explanations for all questions in the True/False section in your B
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Math 108A Winter Quarter 2014 Practice Problems for Midterm No notes or calculators are permitted on this exam. To obtain full credit you must show all work, provide complete proofs, and provide short but complete explanations for all questions in the Tru
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Math 8 Fall Quarter 2013 Practice Problems for Midterm Solutions No notes or calculators are permitted on this exam. To obtain full midterm credit you must show all work, provide complete proofs, and provide short by complete explanations for all questio
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Course: Calculus With Applications I
MATH 3A - PRACTICE FIRST MIDTERM EXAM Answers Version A 1. Version B 4 5 1. 1 8 a) 1, 2 2. a = 1, b = 4 b) 0, 0 2. 3. 1 cos() 1, use squeeze thm. c) discontinuous at x = 0 3. lim f (x) = 0 = lim f (x) x0 4. x0+ 1 2 b) a) 5. f (0) = 2 and f (3) = 7, use I
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Course: Calculus With Applications I
MATH 3A - PRACTICE FIRST MIDTERM EXAM Spring 2009, Version B Perm #: Discussion time: NAME: No Calculators. Solve each problem in the blue book. Number your solutions according to the corresponding problems. No points will be given for answers with no exp
School: UCSB
Course: Calculus With Applications I
MATH 3A - PRACTICE SECOND MIDTERM EXAM Answers Version A 1. 2x 2x4 +5 y2 y 2. a) 21 m/s Version B 1. tan x tan y 2. b) 2 b) 5 seconds c) 4 3. 0.36 cos x a) | cos x| 3. 3x ln 3 3x+1 b) 4 1 x3 x 2 c) (3x + 2) ln x + x2 + 2 4. 4. 5. 1.25 m3 a) -1 3 m/min
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Course: Calculus With Applications I
MATH 3A - PRACTICE FIRST MIDTERM EXAM Spring 2009, Version A Perm #: Discussion time: NAME: No Calculators. Solve each problem in the blue book. Number your solutions according to the corresponding problems. No points will be given for answers with no exp
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Course: Calculus With Applications I
MATH 3A - PRACTICE FINAL EXAM Answers Version A 1. 2. ey + 3y cos x 3 sin x xey a) D = cfw_x = 0, x = 1 y = 0 hor. asympt, x = 1 vert. asympt. Version B 1. 2. 3. 2:12 pm 4. a) D = (, 1) (0, ) 1 y = 2 hor. asympt, no vert. asympt. b) x > 0 increasing, x <
School: UCSB
Course: Calculus With Applications I
MATH 3A - PRACTICE SECOND MIDTERM EXAM Spring 2009, Version A Perm #: Discussion time: NAME: No Calculators. Solve each problem in the blue book. Number your solutions according to the corresponding problems. No points will be given for answers with no ex
School: UCSB
Course: Calculus With Applications I
MATH 3A - PRACTICE FINAL EXAM Spring 2009, Version B Perm #: Discussion time: NAME: No Calculators. Solve each problem in the blue book. Number your solutions according to the corresponding problems. No points will be given for answers with no explanation
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Course: Calculus With Applications I
MATH 3A - PRACTICE FINAL EXAM Spring 2009, Version A Perm #: Discussion time: NAME: No Calculators. Solve each problem in the blue book. Number your solutions according to the corresponding problems. No points will be given for answers with no explanation
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Course: Calculus With Applications I
MATH 3A - PRACTICE SECOND MIDTERM EXAM Spring 2009, Version B Discussion time: Perm #: NAME: No Calculators. Solve each problem in the blue book. Number your solutions according to the corresponding problems. No points will be given for answers with no ex
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Course: Calculus For Social And Life Sciences
MATH 34B - PRACTICE FINAL EXAM Spring 2009, Version B Perm #: Discussion time: NAME: No Calculators. Solve each problem in the blue book. Number your solutions according to the corresponding problems. No points will be given for answers with no explanatio
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Course: Calculus For Social And Life Sciences
MATH 34B - PRACTICE FIRST MIDTERM EXAM Spring 2009, Version B Perm #: Discussion time: NAME: No Calculators. Solve each problem in the blue book. Number your solutions according to the corresponding problems. No points will be given for answers with no ex
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Course: Calculus For Social And Life Sciences
MATH 34B - PRACTICE SECOND MIDTERM EXAM Answers Version A 1. 2. Version B 5 ln(5/2) ln(3/2) 7 1. 200e 5 811 2 2 a) 2 cos x sin x 2. (x 1)2 ex 3 x +C c) cos 3 b) 3. t3 +C 3 1 t3 b) 3e 3 t + + 4 3 1 a) 3e 3 t + 4. x = 1, min; x = 1, max 5. ln 6 hours ln 81
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Course: Calculus For Social And Life Sciences
MATH 34B - PRACTICE FIRST MIDTERM EXAM Spring 2009, Version A Perm #: Discussion time: NAME: No Calculators. Solve each problem in the blue book. Number your solutions according to the corresponding problems. No points will be given for answers with no ex
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Course: Calculus For Social And Life Sciences
MATH 34B - PRACTICE FINAL EXAM Answers Version A 1. Version B a) 1.5, 1.8 1. z = 6x + 2y 1 b) 8.1 2. 3 a) x2 + e2x ln(2)x + C 2 b) 6x ln(x) + 5x + 9e3x+7 2. c) 2x3 + 14y c) 2(s + t) 6ts 3. a) y = ln b) 400 9 4 9 4 y 5. a) t2 + 3t 4 b) 10.5 m 3. 10 4. x =
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Course: Calculus For Social And Life Sciences
MATH 34B - PRACTICE FIRST MIDTERM EXAM Answers Version A 1. 37 1 3 2. 4 3 b) 12 a) 102 t% b) after 100 days 16 4. 3 5. 1. 595 m 6 a) 21 b) 4 c) 1 + 6 x + 8x a) 2 c) 2x + 3 x 3. Version B 2. 1 hour 40 minutes 3. 25 2 4. between 62.6 and 69.2 liters 5. 3 ye
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Course: Calculus For Social And Life Sciences
MATH 34B - PRACTICE FINAL EXAM Spring 2009, Version A Perm #: Discussion time: NAME: No Calculators. Solve each problem in the blue book. Number your solutions according to the corresponding problems. No points will be given for answers with no explanatio
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Course: Calculus For Social And Life Sciences
MATH 34B - PRACTICE SECOND MIDTERM EXAM Spring 2009, Version B Perm #: Discussion time: NAME: No Calculators. Solve each problem in the blue book. Number your solutions according to the corresponding problems. No points will be given for answers with no e
School: UCSB
Course: Calculus For Social And Life Sciences
MATH 34B - PRACTICE SECOND MIDTERM EXAM Spring 2009, Version A Perm #: Discussion time: NAME: No Calculators. Solve each problem in the blue book. Number your solutions according to the corresponding problems. No points will be given for answers with no e
School: UCSB
Course: Vector Calculus I
Math 5B, Midterm 1 Review Problems - Solutions Fall 2006 1. Consider the two planes P1 , dened by the equation 2x y + z = 1, and P2 , given by the equation x + y z = 3. (a) Find parametric equations for the line that is the intersection P1 P2 of the plane
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Course: Vector Calculus I
Math 5B, Solutions to Final Review Problems Fall 2006 1 1 y 1. Integrate 0 sin x dx dy. x Solution. Since the integral sin x dx is too hard, we change the order of integration x so that we integrate with respect to y rst. This double integral is taken ove
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Course: Vector Calculus I
Math 5B, Midterm 2 Review Problems Fall 2006 1. (a) Convert the point (1, 1, 1) from rectangular to cylindrical coordinates. Solution. (It helps to draw pictures!) To convert to cylindrical coordinates, all 2 2 we need to do is change the x, y-coordinates
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Course: Vector Calculus I
Math 5B Midterm Exam Spring 2006 Your name: Your perm.: Your signature: Scores: 1. 2. 3. 4. 5. Total: (out of 100) Note: 12 extra credit points are included. 1 1. (22 points) 1)(15 points) Find the dierential dw of the function w = f (x, y, z) = x2 yz + x
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Course: Vector Calculus I
Math 5B - Midterm 1 Solutions 1. (a) Find parametric equations for the line that passes through the point (2, 0, 1) and is perpendicular to the plane with equation 4x y 2z = 1. Solution. The direction vector for this line is v = (4, 1, 2) and it must pass
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Course: Vector Calculus I
Math 5B, Midterm 2 Solutions Fall 2006 1. Suppose u = cos x + y and v = sin y x. Find x u v and y . v u (x,y) Solution. The Jacobian matrix (u,v) is simply the inverse of the Jacobian matrix sin x 1 (u,v) = . The determinant of this matrix is sin x cos y
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Course: Vector Calculus I
Math 5B, Midterm 2 Review Problems Fall 2006 1. (a) Convert the point (1, 1, 1) from rectangular to cylindrical coordinates. (b) Convert (2, /2, 2/3) from spherical to rectangular coordinates. 2. Suppose z and w are functions of x and y given by the equat
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Course: Vector Calculus II
Math 5C, Final Exam Review Problems Winter 2007 The Final Exam will cover material from the entire course, including several problems on vector calculus, and several on sequences and series. You should thus also review old homework problems from Chapters
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Course: Vector Calculus II
Math 5C, Midterm 1 Review Problems Winter 2007 1. Compute C yz dx + 2x dy y dz where C is the straight line path from (1, 2, 1) to (1, 3, 0). 2. Find the surface area of the surface S, which is parametrized by x(u, v) = u v y(u, v) = u + v (u, v) = z(u, v
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Course: Vector Calculus II
Midterm 1 Solutions 1. [10 Points] Evaluate C ydx + ydy + xdz where C is the straight line path from (3, 1, 2) to (2, 1, 1). Does this integral depend on the path from (3, 1, 2) to (2, 1, 1)? Explain. Solution: Parametrize the curve by x = 3 t, y = 1 + 2t
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Course: Vector Calculus II
Solutions to Final Exam Review Problems Math 5C, Winter 2007 1. Let f (x) = 1 . 4+x (a) Find the Maclaurin series for f (x), and compute its radius of convergence. 1 (1)n n 1 n = (x/4) = Solution. f (x) = x . Since the 4(1 (x/4) 4 n=0 4n+1 n=0 innite seri
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Course: Vector Calculus II
Math 5C, Solutions to Midterm 1 Review Problems Winter 2007 1. Compute C yz dx + 2x dy y dz where C is the straight line path from (1, 2, 1) to (1, 3, 0). Solution. The straight line path between the two points has parametric equations (x, y, z) = (2t + 1
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Course: Vector Calculus II
Math 5C, Midterm 2 Review Problems Winter 2007 1. The rst several terms of a sequence are given. Find a formula in terms of n for the nth term of the sequence (be sure to say what value n starts at). Then nd the limit of the sequence as n tends towards in
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Course: Transitions To Higher Mathematics
Math 8 - Midterm 1 Solutions October 19, 2007 1. (12 pts) Consider the proposition R If I go surng or take a nap, then I will not go surng or I will not take a nap. (a) (2 pts) Express this proposition symbolically in terms of propositional variables P an
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Course: Transitions To Higher Mathematics
Solutions to Midterm 2 Review Problems Math 8, Fall 2007 1. Write the following sets in the form a) cfw_x S | P (x) and the form b) cfw_f (x) | x S . (i) cfw_1, 2, 3, 2, 5, . . . (ii) cfw_11, 21, 31, 41, . . . Solution. (i) a) cfw_x R | x2 N x > 0; b) cf
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Course: Transitions To Higher Mathematics
Math 8 - Midterm 2 Solutions Fall, 2007 1. Give an example of a function with the stated property, or briey explain why no such function can exist. (a) A surjective function f : A B that is not injective. (Please also specify the sets A and B .) Solution.
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Course: Transitions To Higher Mathematics
Math 8 - Final Exam Review Problems Winter 2007 While the nal exam will focus on topics covered since the second midterm (Ch. 2.10, 3.2-3.3, 4.1-4.3), you will still be expected to know all the material from chapters 1 and 2 as well. The MOST IMPORTANT to
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Homework 1 Solutions 1. (#1.1.2 in Strauss) Which of the following operators are linear? (a) Lu = ux + xuy (b) Lu = ux + uuy (c) Lu = ux + u2 y (d) Lu = ux + uy + 1 (e) Lu = 1 + x2 (cos y )ux + uyxy [arctan(x/y )]u Solution: (a) Linear. (b) Nonlinear the
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Course: 117
Homework 2 Hctor Guillermo Cullar R e e os February 2, 2006 12.12 Let D be a nonempty set and suppose that f : D R and g : D R. Dene the function f + g : D R by (f + g)(x) = f (x) + g(x). (a) If f (D) and g(D) are bounded above, then prove that (f
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Homework 5 Math 104A, Fall 2010 Due on Tuesday, November 9th, 2010 1. Given xi , i = 0, 1, . . . , n, consider the Lagrange polynomials Ln,j for j = 0, 1, . . . , n. Prove that n Ln,j (x) = 1 for all x R. j =0 2. The following data is taken from a polynom
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Course: Graph Theory
HOMEWORK 3 SOLUTIONS (1) Show that for each n N the complete graph Kn is a contraction of Kn,n . Solution: We describe the process for several small values of n. In this way, we can discern the inductive step. Clearly, K1 , which is just one vertex, is a
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Robin Meister Homework Set HW11 due 10/25/2013 at 05:00am PDT MATH 34A Fall 2012 !Correct Answers: Cooper 2.2.11 (5)/(70-5-55) (5)/(v-55-5) 1. (1 pt) To do this question you need to know that speed is distance travelled divided by time taken. Three obje
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Robin Meister Homework Set HW13 due 10/30/2013 at 06:00am PDT MATH 34A Fall 2012 a)7100 1. (1 pt) Cooper 7.13.6 Use logs and antilogs to perform the following calculations. DO NOT JUST MULTIPLY THE WAY YOU LEARNED IN 3rd GRADE OR USE A CALCULATOR! a)335 4
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Robin Meister Homework Set HW18 due 11/13/2013 at 05:00am PST MATH 34A Fall 2012 !Correct Answers: 2*50-52 Cooper 3.2.15 1. (1 pt) (a) A cube has surface area 294m2 . What is the volume of the cube? Volume of Cube= m3 Cooper 3.2.32 5. (1 pt) The number N
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Robin Meister Homework Set HW8 due 10/18/2013 at 05:00am PDT MATH 34A Fall 2012 Cooper 1.5.32 1. (1 pt) If f is the focal length of a lens and u is the distance of an object from the lens and v is the distance of the image from the lens then 1 1 1 + =. u
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Robin Meister Homework Set HW7 due 10/16/2013 at 05:00am PDT MATH 34A Fall 2012 (a) How far apart are they after 1 hour? 1. (1 pt) Cooper 1.5.9 (b) How far apart are they after t hours? (c) When are they 4160 miles apart? a) miles Solve (x 10)(x 20)(x 30)
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October 15, 2011 13:47 ast Sheet number 3 Page number xx cyan magenta yellow black October 10, 2011 15:10 frs Sheet number 3 Page number iii cyan magenta yellow black 10 th EDITION David Henderson/Getty Images CALCULUS EARLY TRANSCENDENTALS HOWARD ANTON I
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Course: Math 4A
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(sample solutions) Math 117, W 14, HW#7 P.105, #17. Suppose : D R with f (x) 0 for all x D. Show that, if f is f continuous at x0 , then f is continuous at x0 . Proof. Dene g : [0, ) R to be the square root function: g(x) = x for x [0, ). By Exercise 6,
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(sample solutions) Math 117, W 14, HW#4 P.57, #35. Suppose x is an accumulation point of cfw_an | n N. Show that there is a subsequence of (an )n=1 that onverges to x. Proof. For abbreviations sake, set S = cfw_an | n N. 1 We claim that there are positive
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(sample solutions) Math 117, W 14, HW#6 P.104, #1. Dene f : R R by f (x) = 3x2 2x + 1. Show that f is continuous at 2. [This problem was assigned to be done without using the theorems from Section 3.2.] Proof. Let > 0. Take = mincfw_1, 13 > 0. If x R wit
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Math 117, W 14, HW#1 (sample solutions) P.54, #3. Suppose x R and of its members. > 0. Prove that (x , x + ) is a neighborhood of each Proof. Let y (x , x + ). Then x < y < x + , and so the dierences y (x ) and (x + ) y are both positive. Set = mincfw_y (
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(sample solutions) Math 117, W 14, HW#3 P.56, #25. Suppose (an )n=1 and (bn )n=1 are sequences such that (an )n=1 and (an + bn )n=1 converge. Prove that (bn )n=1 converges. First proof. Since the constant sequence (1)n=1 and the given sequence (an )n=1 ar
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(sample solutions) Math 117, W 14, HW#2 P.55, #21. Determine the accumulation points of the set cfw_2n + 1 k | n, k N. 1 Answer. Set S = cfw_2n + k | n, k N. Claim: The accumulation points of S are precisely the numbers 2n where n is any positive integer.
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(sample solutions) Math 117, W 14, HW#5 P.79, #11. Suppose that f, g, h : D R where x0 is an accumulation point of D, that f (x) g(x) h(x) for all x D, and that f and h have limits at x0 . Assume that lim f (x) = lim h(x). Prove that g has a limit at x0 a
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108A 10th Homework Solutions March 13, 2014 1 Problem 39 Proof. We argue by contradiction and suppose T has at least 6 distinct eigenvalue. By dimension formula (1.1) dim(R7 ) = dim(N ull(T ) + dim(Image(T ) Then dim(Image(T ) = 4. By theorem 5.6 of text
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108A 6th Homework Solutions February 18, 2014 1 Problem 24 Solution: (i) We prove P n = p for all integers n by induction. For n = 1, 2 it is true. Suppose the statement is true for n = k. P k+1 = P P k = P P = P . So the statement is also true for n = k
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108A 9th Homework Solutions March 11, 2014 1 Problem 34 Proof. Since T is invertible, then 0 is not a eigenvalue of T since if not, T will have a nontrivial kernel. Suppose = 0 is a eigenvalue of T, namely, v = 0, such that T v = v. Since A is invertible,
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108A 8th Homework Solutions March 4, 2014 1 Problem 30 Solution: (1) cos sin (1.1) sin cos A rotation with some angle (0, ) (Since this type rotation dont have 2 real eigenvalue, and it hasnt any eigenvector, hence no 1 dimensional invariant subspace
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108A Second Homework Solutions February 25, 2014 1 Problem 27 Proof. We know that 3 5 = 1 in Z7 . 3x2 + 2 = 0. Then 3x2 = 2 = 5, hence 5 3x2 = 5 5. This means x2 = 4. Plug in all 7 elments in Z7 , the solutions are 2 and 5. 2 Problem 28 Proof. We assume t
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108A 4th Homework Solutions February 3, 2014 1 Problem 14 Solution: This is a linear map. Proof. Given any two vectors (a1 , b1 , c1 , d1 ), (a2 , b2 , c2 , d2 ) R4 , L(a1 , b1 , c1 , d1 ) + (a2 , b2 , c2 , d2 ) = L(a1 + a2 , b1 + b2 , c1 + c2 , d1 + d2 )
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1 Problem 1 Solution: Notice that (a + ib)(a ib) = a2 + b2 . Then we have (1.1) r + is = 1 a ib a ib a b = = 2 = 2 +i 2 a + ib (a ib)(a + ib) a + b2 a + b2 a + b2 So we have r = 2 aib a2 +b2 and s = b a2 +b2 Problem 2 Solution: By the denition, if z = a
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108A Second Homework Solutions January 21, 2014 1 Problem 5 Proof. Let Q[x] = cfw_p(x) = an xn +an1 xn1 + +a1 x+a0 : n N, an , , a0 Q, an = 0 and V = cfw_p(x) Q[x] : p( 2) = 0. We want to show V is a vector space. First we show Q[x] is vector space. The
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108A 3rd Homework Solutions February 12, 2014 1 Problem 9 (i) Proof. Let P [x] = cfw_p(x) = an xn +an1 xn1 + +a1 x+a0 : n N, an , , a0 R, an = 0 and W = cfw_p(x) P [x] : deg p(x) < 5 and p(x) = (x5)g(x) f or some g(x) R[x]. We want to show W is a vector
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Math 8 Fall Quarter 2013 Homework 1 Solutions 1. (i) Prove that the product of two consecutive integers is an even number. Solution: Let a, a + 1 denote a pair of consecutive integers. Suppose a is even. Then there exists k Z such that a = 2k. Thus a(a +
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Math 8 Fall Quarter 2013 Homework 4 Soutions 1. Draw on the complex plane the complex numbers z for which: (i) 1 Re(z) 1 (ii) arg(z) = 3 4 (iii) arg(z) 4 4 (iv) |z| 2 and arg(z) 0. Solution See Figures 1(i)-(iv). Figure 1: (i) The complex numbers z for w
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Math 8 Fall Quarter 2013 Homework 3 Solutions 1. For x, y real numbers, prove the following: (i) xy > 0 x > 0 y (ii) If x > y and u > v, then x+u > y +v. Investigate if there is a similar rule for the subtraction. (iii) If x > y > 0 and u > v > 0, then xu
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Math 8 Fall Quarter 2013 Homework 2 Solutions Number Systems 1. Find geometrically on the line of real numbers the points that represent the following numbers: (i) 2/5, (ii) 10. Solution: For nding the precise positions of rational numbers on the real li
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Course: Introduction To Partial Differential Equations
5.3 ORTHOGONALITY AND GENERAL FOURIER SERIES MATH 124B Solution Key HW 02 5.3 ORTHOGONALITY AND GENERAL FOURIER SERIES 1. (a) Find the real vectors that are orthogonal to the given vectors (1, 1, 1) and (1, 1, 0). (b) Choosing an answer to (a), expand the
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I N E A STEPHEN H. FRIEDBERG R L ARNOLD J. INSEL G E B R A LAWRENCE E. SPENCE List Aj A'1 Al A* Al3 A' (A\B) o if Symbols t he ij-th e ntry of the matrix A t he inverse of the matrix A page 9 page 100 t he pseudoinverse of the matrix A page 414 page 331 t
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Solutions to Linear Algebra, Fourth Edition, Stephen H. Friedberg, Arnold J. Insel, Lawrence E. Spence Jephian Lin, Shia Su, Zazastone Lai July 27, 2011 Copyright 2011 Chin-Hung Lin. Permission is granted to copy, distribute and/or modify this document un
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Course: Transitions To Higher Mathematics
Math 8 - Solutions to Home Work 2 Due: October 11, 2007 1. Every/Only. Sometimes sentences with the words only and every can be conditional statements in disguise. For example, Every even number is a multiple of two. can be rephrased as If a number is eve
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Course: Transitions To Higher Mathematics
Math 8 - Homework #3 Solutions October 18, 2007 For exercises 1-3, do the following: (a) Rewrite the given proposition as a conditional (if-then) statement. (b) Prove the proposition or give a counterexample. (c) If you prove it, say whether your proof is
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Course: Transitions To Higher Mathematics
Math 8 - Homework #4 Solutions Fall, 2007 1. Express each of the following statements using sets. Your answers should be of the form [something] (or ) [some set]. / (a) x is a nonnegative integer that is smaller than 5. (b) Either a or b equals 1. (c) Nei
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Course: Advanced Linear Algebra
Math 108B - Home Work # 1 Solutions 1. For T to have the matrix 10 0 1 00 with respect to a basis cfw_u1 , u2 of R2 and a basis cfw_v1 , v2 , v3 for R3 , means simply that T u1 = v1 and T u2 = v2 . Hence cfw_u1 , u2 can remain the standard basis, and t
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Course: Advanced Linear Algebra
Math 108B - Home Work # 3 Solutions 1. LADR Problems. 10. We have |1| = 1 0 12 dx = 1, so we can take e1 = 1. Now let 1 u2 = x x, e1 e1 = x x, 1 1 = x 0 xdx = x 1/2. 12x 12/2. Now let Since |u2 | = 1 (x 0 1/2)2 dx = 1/12, we let e2 = u3 = x2 x2 , e1 e1
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Course: Advanced Linear Algebra
Math 108B - Home Work # 6 Solutions 1. If A is an n n upper-triangular matrix (i.e., Aij = 0 for all i > j), show that det A = n Aii . i=1 Solution. As done in class, we can compute the determinant of A by simplifying the wedge product of the columns of A
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Course: Advanced Linear Algebra
Math 108B - Home Work # 5 Solutions 1. LADR Problems, p. 159-160: 11. Let T be a normal operator on the complex inner-product space V . By the spectral theorem there is an orthonormal basis cfw_e1 , . . . , en of V consisting of eigenvectors T . for If T
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Course: Advanced Linear Algebra
Math 108B - Home Work # 4 Solutions LADR Problems p. 125 24. Notice that (p) = p(1/2) is a linear functional on P2 (R). Thus we follow the idea 1 of the proof of 6.45 to nd a polynomial q P2 (R) such that (p) = p, q = 0 p(x)q(x) dx for all p. need an orth
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Robin Meister Homework Set HW10 due 10/23/2013 at 05:00am PDT MATH 34A Fall 2012 Cooper 5.1.5 Cooper 6.2.8 1. (1 pt) Find the following limit (Hint: simplify the fraction before thinking about limits.) What would happen if you just plug in h=0 before doin
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Robin Meister Homework Set HW6 due 10/15/2013 at 06:00am PDT MATH 34A Fall 2012 Cooper 3.2.37 Cooper 6.1.4 1. (1 pt) Which number gives the same result when you subtract 9 as when you divide by 9? 6. (1 pt) Find the equation of the line through (2,a) and
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Robin Meister Homework Set HW5 due 10/09/2013 at 05:00am PDT MATH 34A Fall 2012 Cooper 1.5.22 Cooper 3.2.1 1. (1 pt) Solve: 5. (1 pt) A rectangular eld is to have an area of 25m2 and is to be surrounded by a fence. The cost C of the fence is 10 dollars pe
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Robin Meister Homework Set HW4 due 10/07/2013 at 05:00am PDT MATH 34A Fall 2012 !- 1. (1 pt) Cooper 1.2.3 Correct Answers: 2*9-3 3-9 Chose the correct expansions for the following equations: (a) (x + y) (x + y) (x + y) (b) (1 + p + p2 ) (1 p) A. a) x3
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Robin Meister Homework Set HW2 due 10/02/2013 at 05:00am PDT MATH 34A Fall 2012 100-100*(20/100*7+80/100*7)/(7+7) 1. (1 pt) Cooper 1.2.2 4. (1 pt) Cooper 1.3.10 Expand the following (a) (3x 4y)(2x + 5y) (b) (2x2 3x + 1)(x 1) A. a) 6x2 + 7xy 20y3 b) 2x3
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Robin Meister Homework Set HW19 due 11/15/2013 at 05:00am PST MATH 34A Fall 2012 Cooper 2.2.15 (c) What is the average rate of change of f (t ) between t = 3 and t = 3.1 1. (0 pts) Do question 2.2.15 from your textbook on scratch paper. !- (d) On graph pa
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WeBWorK assignment number HW20 is due 11/18/2013 at 05:00am PST. You can use the Feedback button on each problem page to send e-mail to the professors. Heres the list of the functions which WeBWorK understands. 462000000 Cooper 8.4.2 3. (1 pt) The volume
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Robin Meister Homework Set HW17 due 11/11/2013 at 05:00am PST MATH 34A Fall 2012 For the following problems, enter the upper limit in the rst box, the summand in the second box and the lower limit in the third box. For the summand, enter all multiplicatio
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Robin Meister Homework Set HW16 due 11/08/2013 at 05:00am PST MATH 34A Fall 2012 What is the slope of the graph at x=4.7? [to nd this: draw a line tangent to your graph at that point and work out the slope] Cooper 2.2.29 1. (1 pt) The graph shows the numb
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Math 6B Midterm Review #1 1 Greens Theorem (y + e 1. Compute the line integral x ) dx + (2x + cos(y 2 ) dy where C is the positively oriented boundary C of the region D enclosed by the parabolas y = x2 and x = y 2 . Solution: Here we use (curl form) Q P x
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Course: Calc With Appli 2
This is a study guide for the second Math 3B midterm. It indicates which types of problems you may be expected to answer on the midterm, with instructions on where to nd these topics in the Stewart calculus book. Plenty of examples can be found in the e
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Course: Math 4A
Math 4A Syllabus Winter 2014 Lecture: MWF 10:00 10:50am, MUSIC LLCH Text: Linear Algebra with applications by David C. Lay, Addison-Wesley, 4th Edition. Material to be covered: Chapters 1-6 in the text book. iClicker: You should purchase an iClicker and b
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Math 108a Professor: Padraic Bartlett Syllabus for Math 108a Weeks 1-10 UCSB 2013 Basic Course Information Professor: Padraic Bartlett. Class time/location: MWF 9-9:50, Phelps 3505. Oce hours/location: TTh 2-3pm, South Hall 6516. Additionally, I am tea
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Analysis with an introduction to proof - Math 117 Spring2009 Monday, Wednesday, & Friday, 12:00-12:50pm, South Hall 6635 Instructor: Carlos J. Garc a-Cervera. Oce: South Hall, 6707. Phone: 893-3681. E-mail: cgarcia@math.ucsb.edu URL: http:/www.math.ucsb.e
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Analysis with an introduction to proof - Math 117 Spring2009 Monday, Wednesday, & Friday, 12:00-12:50pm, South Hall 6635 Instructor: Carlos J. Garc a-Cervera. Oce: South Hall, 6707. Phone: 893-3681. E-mail: cgarcia@math.ucsb.edu URL: http:/www.math.ucsb.e
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Intro. to Numerical Analysis - Math 104B Winter 2011 Tuesday & Thursday, 8:00-9:15am, South Hall 6635 Instructor: Carlos J. Garc a-Cervera. Oce: South Hall, 6707. Phone: 893-3681. E-mail: cgarcia@math.ucsb.edu URL: http:/www.math.ucsb.edu/~cgarcia/Courses
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Math 104A, Fall 2010 Intro. to Numerical Analysis Tuesday & Thursday, 9:30-10:45am, 387 101 Instructor: Carlos J. Garc a-Cervera. Oce: South Hall, 6707. Phone: (805) 893-3681. E-mail: cgarcia@math.ucsb.edu URL: http:/www.math.ucsb.edu/~cgarcia/Courses/Mat
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Math 104A, Winter 2009 Intro. to Numerical Analysis Monday, Wednesday, & Friday, 9:00-9:50am, Arts 1426 Instructor: Carlos J. Garc a-Cervera. Oce: South Hall, 6707. Phone: (805) 893-3681. E-mail: cgarcia@math.ucsb.edu URL: http:/www.math.ucsb.edu/~cgarcia
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Math 108B Intro to Linear Algebra Winter 2010 Professor: Kenneth C. Millett Office: 6512 South Hall Office Hours: R 8:30 11:00 Email: millet@math.ucsb.edu Graduate Assistant: Tomas Kabbabe Office: 6432K South Hall Office Hour: W 10:00 11:00 Email: tomas@m
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Course: Numerical Analysis
University of California, Santa Barbara Department of Statistics & Applied Probability PSTAT 120B, Probability & Statistics, Spring 2010 Instructor: Jarad Niemi Email: niemi@pstat.ucsb.edu Course hours: MWF 10:00-10:50am in HFH 1104 TAs: Varvara Kulikova