Announcements Monday, November 19
|
|
- Brittany Barber
- 5 years ago
- Views:
Transcription
1 Announcements Monday, November 19 You should already have the link to view your graded midterm online. Course grades will be curved at the end of the semester. The percentage of A s, B s, and C s to be awarded depends on many factors, and will not be determined until all grades are in. Individual exam grades are not curved. Send regrade requests by tomorrow. WeBWorK 6.6, 7.1, 7.2 are due the Wednesday after Thanksgiving. No more quizzes! My office is Skiles 244 and Rabinoffice hours are: Mondays, 12 1pm; Wednesdays, 1 3pm. (But not this Wednesday.)
2 Section 7.2 Orthogonal Complements
3 Orthogonal Complements Definition Let W be a subspace of R n. Its orthogonal complement is W = { v in R n v w = 0 for all w in W } read W perp. Pictures: W is orthogonal complement A T is transpose The orthogonal complement of a line in R 2 is the perpendicular line. [interactive] W W The orthogonal complement of a line in R 3 is the perpendicular plane. [interactive] W W The orthogonal complement of a plane in R 3 is the perpendicular line. [interactive] W W
4 Poll
5 Orthogonal Complements Basic properties Let W be a subspace of R n. Facts: 1. W is also a subspace of R n 2. (W ) = W 3. dim W + dim W = n 4. If W = Span{v 1, v 2,..., v m}, then W = all vectors orthogonal to each v 1, v 2,..., v m = { x in R n x v i = 0 for all i = 1, 2,..., m } v T 1 = Nul v2 T... vm T
6 Orthogonal Complements Computation 1 Problem: if W = Span 1, 1 1 1, compute W. 1 [interactive] v T 1 Span{v 1, v 2,..., v m} = Nul v2 T.. vm T
7 Orthogonal Complements Row space, column space, null space Definition The row space of an m n matrix A is the span of the rows of A. It is denoted Row A. Equivalently, it is the column space of A T : It is a subspace of R n. Row A = Col A T. We showed before that if A has rows v T 1, v T 2,..., v T m, then Hence we have shown: Fact: (Row A) = Nul A. Span{v 1, v 2,..., v m} = Nul A. Replacing A by A T, and remembering Row A T = Col A: Fact: (Col A) = Nul A T. Using property 2 and taking the orthogonal complements of both sides, we get: Fact: (Nul A) = Row A and Col A = (Nul A T ).
8 Orthogonal Complements Reference sheet Orthogonal Complements of Most of the Subspaces We ve Seen For any vectors v 1, v 2,..., v m: v T 1 Span{v 1, v 2,..., v m} = Nul v2 T.. vm T For any matrix A: Row A = Col A T and (Row A) = Nul A (Col A) = Nul A T Row A = (Nul A) Col A = (Nul A T ) For any other subspace W, first find a basis v 1,..., v m, then use the above trick to compute W = Span{v 1,..., v m}.
9 Section 7.3 Orthogonal Projections
10 Best Approximation Suppose you measure a data point x which you know for theoretical reasons must lie on a subspace W. x x y y W Due to measurement error, though, the measured x is not actually in W. Best approximation: y is the closest point to x on W. How do you know that y is the closest point? The vector from y to x is orthogonal to W : it is in the orthogonal complement W.
11 Orthogonal Decomposition Theorem Every vector x in R n can be written as x = x W + x W for unique vectors x W in W and x W in W. The equation x = x W + x W is called the orthogonal decomposition of x (with respect to W ). The vector x W is the orthogonal projection of x onto W. The vector x W is the closest vector to x on W. x [interactive 1] [interactive 2] x W x W W
12 Orthogonal Decomposition Justification Theorem Every vector x in R n can be written as x = x W + x W for unique vectors x W in W and x W in W. Why?
13 Orthogonal Decomposition Example Let W be the xy-plane in R 3. Then W is the z-axis. 2 x = 1 = x W = x W =. 3 a x = b = x W = x W =. c This is just decomposing a vector into a horizontal component (in the xy-plane) and a vertical component (on the z-axis). x x W x W W [interactive]
14 Orthogonal Decomposition Computation? Problem: Given x and W, how do you compute the decomposition x = x W + x W? Observation: It is enough to compute x W, because x W = x x W.
15 The A T A trick Theorem (The A T A Trick) Let W be a subspace of R n, let v 1, v 2,..., v m be a spanning set for W (e.g., a basis), and let A = v 1 v 2 v m. Then for any x in R n, the matrix equation A T Av = A T x (in the unknown vector v) is consistent, and x W = Av for any solution v. Recipe for Computing x = x W + x W Write W as a column space of a matrix A. Find a solution v of A T Av = A T x (by row reducing). Then x W = Av and x W = x x W.
16 The A T A Trick Example Problem: Compute the orthogonal projection of a vector x = (x 1, x 2, x 3) in R 3 onto the xy-plane.
17 The A T A Trick Another Example Problem: Let 1 x 1 x = 2 W = x 2 in R 3 x1 x 2 + x 3 = 0. 3 x 3 Compute the distance from x to W.
18 The A T A Trick Another Example, Continued Problem: Let 1 x 1 x = 2 W = x 2 in R 3 x1 x 2 + x 3 = 0. 3 x 3 Compute the distance from x to W. [interactive]
19 The A T A trick Proof Theorem (The A T A Trick) Let W be a subspace of R n, let v 1, v 2,..., v m be a spanning set for W (e.g., a basis), and let A = v 1 v 2 v m. Then for any x in R n, the matrix equation A T Av = A T x (in the unknown vector v) is consistent, and x W = Av for any solution v. Proof:
20 Orthogonal Projection onto a Line Problem: Let L = Span{u} be a line in R n and let x be a vector in R n. Compute x L. Projection onto a Line The projection of x onto a line L = Span{u} is x L = u x u u u x L = x x L. x L L x u x L = u x u u u
21 Orthogonal Projection onto a Line Example Problem: Compute the orthogonal projection of x = ( ) 6 4 onto the line L spanned by u = ( 3 2), and find the distance from u to L. ( ) 6 4 ( 3 2) L 10 ( ) [interactive]
22 Summary Let W be a subspace of R n. The orthogonal complement W is the set of all vectors orthogonal to everything in W. We have (W ) = W and dim W + dim W = n. Row A = Col A T, (Row A) = Nul A, Row A = (Nul A), (Col A) = Nul A T, Col A = (Nul A T ). Orthogonal decomposition: any vector x in R n can be written in a unique way as x = x W + x W for x W in W and x W in W. The vector x W is the orthogonal projection of x onto W. The vector x W is the closest point to x in W : it is the best approximation. The distance from x to W is x W. If W = Col A then to compute x W, solve the equation A T Av = A T x; then x W = Av. If W = L = Span{u} is a line then x L = u x u u u.
Announcements Monday, November 19
Announcements Monday, November 19 You should already have the link to view your graded midterm online. Course grades will be curved at the end of the semester. The percentage of A s, B s, and C s to be
More informationAnnouncements Monday, November 20
Announcements Monday, November 20 You already have your midterms! Course grades will be curved at the end of the semester. The percentage of A s, B s, and C s to be awarded depends on many factors, and
More informationAnnouncements Wednesday, November 15
Announcements Wednesday, November 15 The third midterm is on this Friday, November 17. The exam covers 3.1, 3.2, 5.1, 5.2, 5.3, and 5.5. About half the problems will be conceptual, and the other half computational.
More informationAnnouncements Wednesday, November 15
3π 4 Announcements Wednesday, November 15 Reviews today: Recitation Style Solve and discuss Practice problems in groups Preparing for the exam tips and strategies It is not mandatory Eduardo at Culc 141,
More informationAnnouncements Monday, November 26
Announcements Monday, November 26 Please fill out your CIOS survey! WeBWorK 6.6, 7.1, 7.2 are due on Wednesday. No quiz on Friday! But this is the only recitation on chapter 7. My office is Skiles 244
More informationSection 6.1. Inner Product, Length, and Orthogonality
Section 6. Inner Product, Length, and Orthogonality Orientation Almost solve the equation Ax = b Problem: In the real world, data is imperfect. x v u But due to measurement error, the measured x is not
More informationChapter 6. Orthogonality and Least Squares
Chapter 6 Orthogonality and Least Squares Section 6.1 Inner Product, Length, and Orthogonality Orientation Recall: This course is about learning to: Solve the matrix equation Ax = b Solve the matrix equation
More informationAnnouncements Monday, November 26
Announcements Monday, November 26 Please fill out your CIOS survey! WeBWorK 6.6, 7.1, 7.2 are due on Wednesday. No quiz on Friday! But this is the only recitation on chapter 7. My office is Skiles 244
More informationAnnouncements Monday, October 29
Announcements Monday, October 29 WeBWorK on determinents due on Wednesday at :59pm. The quiz on Friday covers 5., 5.2, 5.3. My office is Skiles 244 and Rabinoffice hours are: Mondays, 2 pm; Wednesdays,
More informationAnnouncements Monday, September 17
Announcements Monday, September 17 WeBWorK 3.3, 3.4 are due on Wednesday at 11:59pm. The first midterm is on this Friday, September 21. Midterms happen during recitation. The exam covers through 3.4. About
More informationAnnouncements Wednesday, September 20
Announcements Wednesday, September 20 WeBWorK 1.4, 1.5 are due on Wednesday at 11:59pm. The first midterm is on this Friday, September 22. Midterms happen during recitation. The exam covers through 1.5.
More informationAnnouncements Wednesday, November 01
Announcements Wednesday, November 01 WeBWorK 3.1, 3.2 are due today at 11:59pm. The quiz on Friday covers 3.1, 3.2. My office is Skiles 244. Rabinoffice hours are Monday, 1 3pm and Tuesday, 9 11am. Section
More informationOrthogonal Projection. Hung-yi Lee
Orthogonal Projection Hung-yi Lee Reference Textbook: Chapter 7.3, 7.4 Orthogonal Projection What is Orthogonal Complement What is Orthogonal Projection How to do Orthogonal Projection Application of Orthogonal
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 informationOverview. Motivation for the inner product. Question. Definition
Overview Last time we studied the evolution of a discrete linear dynamical system, and today we begin the final topic of the course (loosely speaking) Today we ll recall the definition and properties of
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 2331 Linear Algebra
6.1 Inner Product, Length & Orthogonality Math 2331 Linear Algebra 6.1 Inner Product, Length & Orthogonality Shang-Huan Chiu Department of Mathematics, University of Houston schiu@math.uh.edu math.uh.edu/
More informationMath 3191 Applied Linear Algebra
Math 191 Applied Linear Algebra Lecture 1: Inner Products, Length, Orthogonality Stephen Billups University of Colorado at Denver Math 191Applied Linear Algebra p.1/ Motivation Not all linear systems have
More informationAnnouncements Wednesday, November 01
Announcements Wednesday, November 01 WeBWorK 3.1, 3.2 are due today at 11:59pm. The quiz on Friday covers 3.1, 3.2. My office is Skiles 244. Rabinoffice hours are Monday, 1 3pm and Tuesday, 9 11am. Section
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 2331 Linear Algebra
4.5 The Dimension of a Vector Space Math 233 Linear Algebra 4.5 The Dimension of a Vector Space Shang-Huan Chiu Department of Mathematics, University of Houston schiu@math.uh.edu math.uh.edu/ schiu/ Shang-Huan
More informationAnnouncements Wednesday, October 04
Announcements Wednesday, October 04 Please fill out the mid-semester survey under Quizzes on Canvas. WeBWorK 1.8, 1.9 are due today at 11:59pm. The quiz on Friday covers 1.7, 1.8, and 1.9. My office is
More informationMarch 27 Math 3260 sec. 56 Spring 2018
March 27 Math 3260 sec. 56 Spring 2018 Section 4.6: Rank Definition: The row space, denoted Row A, of an m n matrix A is the subspace of R n spanned by the rows of A. We now have three vector spaces associated
More informationMATH 1553, SPRING 2018 SAMPLE MIDTERM 2 (VERSION B), 1.7 THROUGH 2.9
MATH 155, SPRING 218 SAMPLE MIDTERM 2 (VERSION B), 1.7 THROUGH 2.9 Name Section 1 2 4 5 Total Please read all instructions carefully before beginning. Each problem is worth 1 points. The maximum score
More informationAnnouncements Monday, November 13
Announcements Monday, November 13 The third midterm is on this Friday, November 17. The exam covers 3.1, 3.2, 5.1, 5.2, 5.3, and 5.5. About half the problems will be conceptual, and the other half computational.
More informationMath 3191 Applied Linear Algebra
Math 9 Applied Linear Algebra Lecture : Orthogonal Projections, Gram-Schmidt Stephen Billups University of Colorado at Denver Math 9Applied Linear Algebra p./ Orthonormal Sets A set of vectors {u, u,...,
More informationAnnouncements Monday, September 18
Announcements Monday, September 18 WeBWorK 1.4, 1.5 are due on Wednesday at 11:59pm. The first midterm is on this Friday, September 22. Midterms happen during recitation. The exam covers through 1.5. About
More informationAnnouncements Wednesday, September 05
Announcements Wednesday, September 05 WeBWorK 2.2, 2.3 due today at 11:59pm. The quiz on Friday coers through 2.3 (last week s material). My office is Skiles 244 and Rabinoffice hours are: Mondays, 12
More informationSection 6.4. The Gram Schmidt Process
Section 6.4 The Gram Schmidt Process Motivation The procedures in 6 start with an orthogonal basis {u, u,..., u m}. Find the B-coordinates of a vector x using dot products: x = m i= x u i u i u i u i Find
More informationAnnouncements Wednesday, November 7
Announcements Wednesday, November 7 The third midterm is on Friday, November 16 That is one week from this Friday The exam covers 45, 51, 52 53, 61, 62, 64, 65 (through today s material) WeBWorK 61, 62
More informationOrthogonal Complements
Orthogonal Complements Definition Let W be a subspace of R n. If a vector z is orthogonal to every vector in W, then z is said to be orthogonal to W. The set of all such vectors z is called the orthogonal
More informationSolutions to Review Problems for Chapter 6 ( ), 7.1
Solutions to Review Problems for Chapter (-, 7 The Final Exam is on Thursday, June,, : AM : AM at NESBITT Final Exam Breakdown Sections % -,7-9,- - % -9,-,7,-,-7 - % -, 7 - % Let u u and v Let x x x x,
More informationAnswer Key for Exam #2
. Use elimination on an augmented matrix: Answer Key for Exam # 4 4 8 4 4 4 The fourth column has no pivot, so x 4 is a free variable. The corresponding system is x + x 4 =, x =, x x 4 = which we solve
More informationMATH 31 - ADDITIONAL PRACTICE PROBLEMS FOR FINAL
MATH 3 - ADDITIONAL PRACTICE PROBLEMS FOR FINAL MAIN TOPICS FOR THE FINAL EXAM:. Vectors. Dot product. Cross product. Geometric applications. 2. Row reduction. Null space, column space, row space, left
More informationAnnouncements Monday, November 13
Announcements Monday, November 13 The third midterm is on this Friday, November 17 The exam covers 31, 32, 51, 52, 53, and 55 About half the problems will be conceptual, and the other half computational
More informationAnnouncements Monday, October 02
Announcements Monday, October 02 Please fill out the mid-semester survey under Quizzes on Canvas WeBWorK 18, 19 are due Wednesday at 11:59pm The quiz on Friday covers 17, 18, and 19 My office is Skiles
More informationAnnouncements Wednesday, November 7
Announcements Wednesday, November 7 The third midterm is on Friday, November 6 That is one week from this Friday The exam covers 45, 5, 52 53, 6, 62, 64, 65 (through today s material) WeBWorK 6, 62 are
More informationLecture 10: Vector Algebra: Orthogonal Basis
Lecture 0: Vector Algebra: Orthogonal Basis Orthogonal Basis of a subspace Computing an orthogonal basis for a subspace using Gram-Schmidt Orthogonalization Process Orthogonal Set Any set of vectors that
More informationMath 51 Midterm 1 July 6, 2016
Math 51 Midterm 1 July 6, 2016 Name: SUID#: Circle your section: Section 01 Section 02 (1:30-2:50PM) (3:00-4:20PM) Complete the following problems. In order to receive full credit, please show all of your
More informationSection 6.2, 6.3 Orthogonal Sets, Orthogonal Projections
Section 6. 6. Orthogonal Sets Orthogonal Projections Main Ideas in these sections: Orthogonal set = A set of mutually orthogonal vectors. OG LI. Orthogonal Projection of y onto u or onto an OG set {u u
More informationThe Fundamental Theorem of Linear Algebra
The Fundamental Theorem of Linear Algebra Nicholas Hoell Contents 1 Prelude: Orthogonal Complements 1 2 The Fundamental Theorem of Linear Algebra 2 2.1 The Diagram........................................
More informationMath 2331 Linear Algebra
6. Orthogonal Projections Math 2 Linear Algebra 6. Orthogonal Projections Jiwen He Department of Mathematics, University of Houston jiwenhe@math.uh.edu math.uh.edu/ jiwenhe/math2 Jiwen He, University of
More informationChapter 3. Directions: For questions 1-11 mark each statement True or False. Justify each answer.
Chapter 3 Directions: For questions 1-11 mark each statement True or False. Justify each answer. 1. (True False) Asking whether the linear system corresponding to an augmented matrix [ a 1 a 2 a 3 b ]
More informationv = v 1 2 +v 2 2. Two successive applications of this idea give the length of the vector v R 3 :
Length, Angle and the Inner Product The length (or norm) of a vector v R 2 (viewed as connecting the origin to a point (v 1,v 2 )) is easily determined by the Pythagorean Theorem and is denoted v : v =
More informationIMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET
IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each
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 6.5. Least Squares Problems
Section 6.5 Least Squares Problems Motivation We now are in a position to solve the motivating problem of this third part of the course: Problem Suppose that Ax = b does not have a solution. What is the
More informationAnnouncements Wednesday, August 30
Announcements Wednesday, August 30 WeBWorK due on Friday at 11:59pm. The first quiz is on Friday, during recitation. It covers through Monday s material. Quizzes mostly test your understanding of the homework.
More informationAnnouncements: Nov 10
Announcements: Nov 10 WebWork 5.3, 5.5 due Wednesday Midterm 3 on Friday Nov 17 Upcoming Office Hours Me: Monday 1-2 and Wednesday 3-4, Skiles 234 Bharat: Tuesday 1:45-2:45, Skiles 230 Qianli: Wednesday
More informationOrthogonality and Least Squares
6 Orthogonality and Least Squares 6.1 INNER PRODUCT, LENGTH, AND ORTHOGONALITY INNER PRODUCT If u and v are vectors in, then we regard u and v as matrices. n 1 n The transpose u T is a 1 n matrix, and
More informationAnnouncements Wednesday, August 30
Announcements Wednesday, August 30 WeBWorK due on Friday at 11:59pm. The first quiz is on Friday, during recitation. It covers through Monday s material. Quizzes mostly test your understanding of the homework.
More informationIMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET
IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each
More informationAnnouncements Wednesday, October 25
Announcements Wednesday, October 25 The midterm will be returned in recitation on Friday. The grade breakdown is posted on Piazza. You can pick it up from me in office hours before then. Keep tabs on your
More informationMath 3191 Applied Linear Algebra
Math 9 Applied Linear Algebra Lecture : Null and Column Spaces Stephen Billups University of Colorado at Denver Math 9Applied Linear Algebra p./8 Announcements Study Guide posted HWK posted Math 9Applied
More informationMath 3C Lecture 25. John Douglas Moore
Math 3C Lecture 25 John Douglas Moore June 1, 2009 Let V be a vector space. A basis for V is a collection of vectors {v 1,..., v k } such that 1. V = Span{v 1,..., v k }, and 2. {v 1,..., v k } are linearly
More informationMATH 20F: LINEAR ALGEBRA LECTURE B00 (T. KEMP)
MATH 20F: LINEAR ALGEBRA LECTURE B00 (T KEMP) Definition 01 If T (x) = Ax is a linear transformation from R n to R m then Nul (T ) = {x R n : T (x) = 0} = Nul (A) Ran (T ) = {Ax R m : x R n } = {b R m
More informationLINEAR ALGEBRA SUMMARY SHEET.
LINEAR ALGEBRA SUMMARY SHEET RADON ROSBOROUGH https://intuitiveexplanationscom/linear-algebra-summary-sheet/ This document is a concise collection of many of the important theorems of linear algebra, organized
More informationWorksheet for Lecture 23 (due December 4) Section 6.1 Inner product, length, and orthogonality
Worksheet for Lecture (due December 4) Name: Section 6 Inner product, length, and orthogonality u Definition Let u = u n product or dot product to be and v = v v n be vectors in R n We define their inner
More informationMidterm solutions. (50 points) 2 (10 points) 3 (10 points) 4 (10 points) 5 (10 points)
Midterm solutions Advanced Linear Algebra (Math 340) Instructor: Jarod Alper April 26, 2017 Name: } {{ } Read all of the following information before starting the exam: You may not consult any outside
More informationMATH 1553, FALL 2018 SAMPLE MIDTERM 2: 3.5 THROUGH 4.4
MATH 553, FALL 28 SAMPLE MIDTERM 2: 3.5 THROUGH 4.4 Name GT Email @gatech.edu Write your section number here: Please read all instructions carefully before beginning. The maximum score on this exam is
More informationAdvanced Linear Algebra Math 4377 / 6308 (Spring 2015) March 5, 2015
Midterm 1 Advanced Linear Algebra Math 4377 / 638 (Spring 215) March 5, 215 2 points 1. Mark each statement True or False. Justify each answer. (If true, cite appropriate facts or theorems. If false, explain
More information1 Last time: determinants
1 Last time: determinants Let n be a positive integer If A is an n n matrix, then its determinant is the number det A = Π(X, A)( 1) inv(x) X S n where S n is the set of n n permutation matrices Π(X, A)
More informationMath 1553 Introduction to Linear Algebra. School of Mathematics Georgia Institute of Technology
Math 1553 Introduction to Linear Algebra School of Mathematics Georgia Institute of Technology Chapter 1 Overview Linear. Algebra. What is Linear Algebra? Linear having to do with lines/planes/etc. For
More informationChapter 6: Orthogonality
Chapter 6: Orthogonality (Last Updated: November 7, 7) These notes are derived primarily from Linear Algebra and its applications by David Lay (4ed). A few theorems have been moved around.. Inner products
More informationSection 3.3. Matrix Equations
Section 3.3 Matrix Equations Matrix Vector the first number is the number of rows the second number is the number of columns Let A be an m n matrix A = v v v n with columns v, v,..., v n Definition The
More informationWe showed that adding a vector to a basis produces a linearly dependent set of vectors; more is true.
Dimension We showed that adding a vector to a basis produces a linearly dependent set of vectors; more is true. Lemma If a vector space V has a basis B containing n vectors, then any set containing more
More informationRank & nullity. Defn. Let T : V W be linear. We define the rank of T to be rank T = dim im T & the nullity of T to be nullt = dim ker T.
Rank & nullity Aim lecture: We further study vector space complements, which is a tool which allows us to decompose linear problems into smaller ones. We give an algorithm for finding complements & an
More informationPRACTICE PROBLEMS FOR THE FINAL
PRACTICE PROBLEMS FOR THE FINAL Here are a slew of practice problems for the final culled from old exams:. Let P be the vector space of polynomials of degree at most. Let B = {, (t ), t + t }. (a) Show
More informationMATH 54 QUIZ I, KYLE MILLER MARCH 1, 2016, 40 MINUTES (5 PAGES) Problem Number Total
MATH 54 QUIZ I, KYLE MILLER MARCH, 206, 40 MINUTES (5 PAGES) Problem Number 2 3 4 Total Score YOUR NAME: SOLUTIONS No calculators, no references, no cheat sheets. Answers without justification will receive
More information2.4 Hilbert Spaces. Outline
2.4 Hilbert Spaces Tom Lewis Spring Semester 2017 Outline Hilbert spaces L 2 ([a, b]) Orthogonality Approximations Definition A Hilbert space is an inner product space which is complete in the norm defined
More informationAnnouncements August 31
Announcements August 31 Homeworks 1.1 and 1.2 are due Friday. The first quiz is on Friday, during recitation. Quizzes mostly test your understanding of the homework. There will generally be a quiz every
More informationMath 22 Fall 2018 Midterm 2
Math 22 Fall 218 Midterm 2 October 23, 218 NAME: SECTION (check one box): Section 1 (S. Allen 12:5) Section 2 (A. Babei 2:1) Instructions: 1. Write your name legibly on this page, and indicate your section
More informationAPPM 2360 Exam 2 Solutions Wednesday, March 9, 2016, 7:00pm 8:30pm
APPM 2360 Exam 2 Solutions Wednesday, March 9, 206, 7:00pm 8:30pm ON THE FRONT OF YOUR BLUEBOOK write: () your name, (2) your student ID number, (3) recitation section (4) your instructor s name, and (5)
More informationLecture 9: Vector Algebra
Lecture 9: Vector Algebra Linear combination of vectors Geometric interpretation Interpreting as Matrix-Vector Multiplication Span of a set of vectors Vector Spaces and Subspaces Linearly Independent/Dependent
More information6. Orthogonality and Least-Squares
Linear Algebra 6. Orthogonality and Least-Squares CSIE NCU 1 6. Orthogonality and Least-Squares 6.1 Inner product, length, and orthogonality. 2 6.2 Orthogonal sets... 8 6.3 Orthogonal projections... 13
More informationMAT 242 CHAPTER 4: SUBSPACES OF R n
MAT 242 CHAPTER 4: SUBSPACES OF R n JOHN QUIGG 1. Subspaces Recall that R n is the set of n 1 matrices, also called vectors, and satisfies the following properties: x + y = y + x x + (y + z) = (x + y)
More informationAnnouncements Wednesday, October 10
Announcements Wednesday, October 10 The second midterm is on Friday, October 19 That is one week from this Friday The exam covers 35, 36, 37, 39, 41, 42, 43, 44 (through today s material) WeBWorK 42, 43
More informationEK102 Linear Algebra PRACTICE PROBLEMS for Final Exam Spring 2016
EK102 Linear Algebra PRACTICE PROBLEMS for Final Exam Spring 2016 Answer the questions in the spaces provided on the question sheets. You must show your work to get credit for your answers. There will
More informationProblem # Max points possible Actual score Total 120
FINAL EXAMINATION - MATH 2121, FALL 2017. Name: ID#: Email: Lecture & Tutorial: Problem # Max points possible Actual score 1 15 2 15 3 10 4 15 5 15 6 15 7 10 8 10 9 15 Total 120 You have 180 minutes to
More informationSTAT 151A: Lab 1. 1 Logistics. 2 Reference. 3 Playing with R: graphics and lm() 4 Random vectors. Billy Fang. 2 September 2017
STAT 151A: Lab 1 Billy Fang 2 September 2017 1 Logistics Billy Fang (blfang@berkeley.edu) Office hours: Monday 9am-11am, Wednesday 10am-12pm, Evans 428 (room changes will be written on the chalkboard)
More informationThis is a closed book exam. No notes or calculators are permitted. We will drop your lowest scoring question for you.
Math 54 Fall 2017 Practice Exam 2 Exam date: 10/31/17 Time Limit: 80 Minutes Name: Student ID: GSI or Section: This exam contains 7 pages (including this cover page) and 7 problems. Problems are printed
More informationDot Products, Transposes, and Orthogonal Projections
Dot Products, Transposes, and Orthogonal Projections David Jekel November 13, 2015 Properties of Dot Products Recall that the dot product or standard inner product on R n is given by x y = x 1 y 1 + +
More informationKevin James. MTHSC 3110 Section 4.3 Linear Independence in Vector Sp
MTHSC 3 Section 4.3 Linear Independence in Vector Spaces; Bases Definition Let V be a vector space and let { v. v 2,..., v p } V. If the only solution to the equation x v + x 2 v 2 + + x p v p = is the
More informationSystems of Linear Equations
Systems of Linear Equations Math 108A: August 21, 2008 John Douglas Moore Our goal in these notes is to explain a few facts regarding linear systems of equations not included in the first few chapters
More informationChapter 6 - Orthogonality
Chapter 6 - Orthogonality Maggie Myers Robert A. van de Geijn The University of Texas at Austin Orthogonality Fall 2009 http://z.cs.utexas.edu/wiki/pla.wiki/ 1 Orthogonal Vectors and Subspaces http://z.cs.utexas.edu/wiki/pla.wiki/
More informationspring, math 204 (mitchell) list of theorems 1 Linear Systems Linear Transformations Matrix Algebra
spring, 2016. math 204 (mitchell) list of theorems 1 Linear Systems THEOREM 1.0.1 (Theorem 1.1). Uniqueness of Reduced Row-Echelon Form THEOREM 1.0.2 (Theorem 1.2). Existence and Uniqueness Theorem THEOREM
More informationShorts
Math 45 - Midterm Thursday, October 3, 4 Circle your section: Philipp Hieronymi pm 3pm Armin Straub 9am am Name: NetID: UIN: Problem. [ point] Write down the number of your discussion section (for instance,
More information(i) [7 points] Compute the determinant of the following matrix using cofactor expansion.
Question (i) 7 points] Compute the determinant of the following matrix using cofactor expansion 2 4 2 4 2 Solution: Expand down the second column, since it has the most zeros We get 2 4 determinant = +det
More information1 Last time: inverses
MATH Linear algebra (Fall 8) Lecture 8 Last time: inverses The following all mean the same thing for a function f : X Y : f is invertible f is one-to-one and onto 3 For each b Y there is exactly one a
More informationSection Instructors: by now you should be scheduled into one of the following Sections:
MAT188H1F LINEAR ALGEBRA: Syllabus for Fall 2018 as of October 26, 2018 2018-2019 Calendar Description: This course covers systems of linear equations and Gaussian elimination, applications; vectors in
More informationMTH 2310, FALL Introduction
MTH 2310, FALL 2011 SECTION 6.2: ORTHOGONAL SETS Homework Problems: 1, 5, 9, 13, 17, 21, 23 1, 27, 29, 35 1. Introduction We have discussed previously the benefits of having a set of vectors that is linearly
More informationMATH 1553, JANKOWSKI MIDTERM 2, SPRING 2018, LECTURE A
MATH 553, JANKOWSKI MIDTERM 2, SPRING 28, LECTURE A Name GT Email @gatech.edu Write your section number here: Please read all instructions carefully before beginning. Please leave your GT ID card on your
More informationUniversity of Ottawa
University of Ottawa Department of Mathematics and Statistics MAT 30B: Mathematical Methods II Instructor: Alistair Savage Second Midterm Test Solutions White Version 3 March 0 Surname First Name Student
More informationLecture 20: 6.1 Inner Products
Lecture 0: 6.1 Inner Products Wei-Ta Chu 011/1/5 Definition An inner product on a real vector space V is a function that associates a real number u, v with each pair of vectors u and v in V in such a way
More informationI. Multiple Choice Questions (Answer any eight)
Name of the student : Roll No : CS65: Linear Algebra and Random Processes Exam - Course Instructor : Prashanth L.A. Date : Sep-24, 27 Duration : 5 minutes INSTRUCTIONS: The test will be evaluated ONLY
More informationMath 200 A and B: Linear Algebra Spring Term 2007 Course Description
Math 200 A and B: Linear Algebra Spring Term 2007 Course Description February 25, 2007 Instructor: John Schmitt Warner 311, Ext. 5952 jschmitt@middlebury.edu Office Hours: Monday, Wednesday 11am-12pm,
More informationMath 21b: Linear Algebra Spring 2018
Math b: Linear Algebra Spring 08 Homework 8: Basis This homework is due on Wednesday, February 4, respectively on Thursday, February 5, 08. Which of the following sets are linear spaces? Check in each
More informationBasic Linear Algebra Ideas. We studied linear differential equations earlier and we noted that if one has a homogeneous linear differential equation
Math 3CI Basic Linear Algebra Ideas We studied linear differential equations earlier and we noted that if one has a homogeneous linear differential equation ( ) y (n) + f n 1 y (n 1) + + f 2 y + f 1 y
More information2018 Fall 2210Q Section 013 Midterm Exam II Solution
08 Fall 0Q Section 0 Midterm Exam II Solution True or False questions points 0 0 points) ) Let A be an n n matrix. If the equation Ax b has at least one solution for each b R n, then the solution is unique
More informationMath 290, Midterm II-key
Math 290, Midterm II-key Name (Print): (first) Signature: (last) The following rules apply: There are a total of 20 points on this 50 minutes exam. This contains 7 pages (including this cover page) and
More information