MAT 419 Lecture Notes Transcribed by Eowyn Cenek 6/1/2012
|
|
- Damian Ryan
- 6 years ago
- Views:
Transcription
1 (Homework 1: Chapter 1: Exercises 1-7, 9, 11, 19, due Monday June 11th See also the course website for lectures, assignments, etc) Note: today s lecture is primarily about definitions Lots of definitions We will use these in the next sections 1 Vectors in R n When working with functions of several variables, we consider the input as a n-vector, where n refers to the dimension of the space we are working (n = the number of variables) Thus x R n is defined as x 1 x 2 x = = (x 1, x 2,, x n ) x n Given two or more vectors, we define the following vector space operations: a) Addition: x + y = (x 1 + y 1, x 2 + y 2,, x n + y n ) b) Scalar multiplication: αx = (αx 1, αx 2,, αx n ) c) Dot product: x y = x 1 y 1 + x 2 y x n y n (Also known as the inner product) d) Norm: x = x x x 2 n = (x x) 1/2 (The length of the vector) The inner product and norm operations have several important properties Defn Properties of the inner product, given two vectors x and y and a constant α: x y = y x (x + y)z = xz + yz (αx)y = α(xy) Defn Properties of the norm, given two vectors x and y and a constant α: x 0 and x = 0 if and only if x = 0 αx = α x x + y x + y (Known as the triangle inequality The length of the longest side is never longer than the sum of the lengths of the shorter sides) x y x y (Known as the Cauchy-Schwarz 1 inequality) 1 And sometimes known as the Cauchy-Schwarz-Buniakowsky inequality p 1 of 5
2 We also need to define the distance between two vectors In R, the difference between two variables is x 1 x 2 In R 2 we use the Euclidean distance between a = (x a, y a ) and b = (x b, y b ), or φ(a, b) = (x a x b ) 2 + (y a y b ) 2 We can define the distance for any positive dimension n: Defn The distance between two vectors x and y in R n is defined as ( n ) 1/2 φ(x, y) = x y = (x i y i ) 2 Lastly we need the definition of an n-dimensional ball to define open and closed sets: define the ball i=1 B(x, r) = {y R n ; φ(x, y) < r} (This is an open ball; the boundary is not included) Given a set D R n, we define x as an interior point if there is some radius r > 0 such that B(x, r) D (In other words, we can draw a ball very small if necessary around x so that the entire ball is contained inside D Points on the boundary of D, for instance, can never be interior points) A set D R n is open if it is equal to its own interior D 0, where D 0 is the set of all interior points of D Ex Examples of open sets: (0, 1) R is open {(x, y); x 2 + y 2 < 1} R 2 is open (and is the unit circle centered at 0) A set D R n is closed if its complement D c is open Ex Examples of closed sets: [0, 1] R is closed {(x, y); x 2 + y 2 1} R 2 is closed (and is the unit circle centered at 0, but now includes the boundary) Typically open sets involve strict inequalities, and closed sets involve or (Odd as it sounds, R n is both open and closed, since it both contains all its interior points, and its complement is the empty set which is technically also open This is the only n-dimensional set which is both open and closed) 2 Functions of Several Variables Now that we have defined vectors, we can move on to functions that take a vector as an input Let f : D R n R n be a function Then x D is a p 2 of 5
3 a) global minimizer of f on D if f(x ) f(x) for all x D b) strict global minimizer of f on D if f(x ) < f(x) for all x D and x x c) local minimizer of f on D if f(x ) f(x) for x B(x, r) for some r > 0 d) strict local minimizer of f on D if f(x ) < f(x) for x B(x, r) for some r > 0, and x x e) x is a critical point if f x i (x ) exists and is equal to 0 for i = 1, 2,, n (Note that the first four conditions are essentially identical to the definitions for a single variable function Condition e) is comparable, but now we require that all the partial derivatives be zero) Theorem (Fermat s Theorem 2 ) If f is differentiable on D R n, and x D 0, and x is a local minimizer or maximizer of f, then x also has to be a critical point of f (This is important because it tells us that every minimum or maximum is a critical point, rather than the converse that every critical point can be a minimum or maximum So if we find all the critical points, we are guaranteed to have all the minimizers and maximizers, and possibly a few extra points) Note that (( ) ( ) ( )) f f f f(x ) =,, x 1 x 2 x n and x is a critical point if and only if f(x ) = 0 ( f(x is the gradient of f at x) 21 Returning to the Taylor Expansion Consider the Taylor Theorem for 1 or more dimensions n = 1 f(x) = f(x ) + f (x )(x x ) f (z)(x x ) 2 for some z between x, x n > 1 f(x) = f(x ) + f (x ) (x x ) (x x ) H f (z)(x x ) where H f (z) is the Hessian n n matrix defined so that 2 One of the smaller ones H f (z) i,j = 2 f x i x j p 3 of 5
4 More specifically, the Hessian looks like 2 f x f H f (x) = x 2 x 1 2 f x n x 1 x 1 x 2 x 1 x n x 2 2 x 2 x n x n x 2 x 2 n Note that H f (z) is a symmetric matrix since = x j x i x i x j 22 Notes on Linear Algebra Let A be an n n symmetric matrix A quadratic form Q A (y) : R n R n is defined by Q A (y) = y (Ay) = n n a ij y i y j i=1 j=1 Ex Consider the function f(x, y, z) = x 2 y 2 + 4z 2 2xy + 4yz We calculate the gradient as ( f f(x, y, z) = x, f y, f ) z = (2x 2y, 2y 2x + 4z, 8z + 4y) H f (x, y, z) = x 2x y Q Hf (x, y, z) = (x, y, z) H f (x, y, z) = (x, y, z) y = (x, y, z) 2y 2x + 4z z 4y + 8z = x(2x y) + y( 2y 2x + 4z) + z(4y + 8z) = 2f(x, y, z) Defn (Positive definite matrix) An n n symmetric matrix A and its quadratic form Q A (y) = y (Ay) is positive definite if Q A (y) > 0 for all nonzero vectors y 0 positive semi-definite if Q A (y) 0 for all vectors y negative definite if Q A (y) < 0 for all nonzero vectors y 0 negative semi-definite if Q A (y) 0 for all vectors y p 4 of 5
5 Now consider the Taylor s theorem for multiple dimensions: f(x) = f(x ) + f (x ) (x x ) (x x ) H f (z)(x x ) for some z between x, x Then if f has a critical point at x, so that f(x ) = 0 and f has continuous first and second partial derivatives on R n, then x is a a) global minimizer if H f (x) is positive semi-definite on R n b) strict global minimizer if H f (x) is positive definite on R n c) global maximizer if H f (x) is negative semi-definite on R n d) strict global maximizer if H f (x) is negative definite on R n and so the challenge lies in determining the form of the Hessian (as far as positive/negative (semi) definiteness (It s easier to rule out that a matrix doesn t satisfy the conditions above, than to prove that it does For instance, a positive definite matrix has only positive entries on the diagonal, and a negative definite matrix has only negative entries on the diagonal Thus the quadratic form in the example is neither positive nor negative definite more work would be required to establish if it was positive or negative semi-definite) p 5 of 5
Functions of Several Variables
Jim Lambers MAT 419/519 Summer Session 2011-12 Lecture 2 Notes These notes correspond to Section 1.2 in the text. Functions of Several Variables We now generalize the results from the previous section,
More informationMATH529 Fundamentals of Optimization Unconstrained Optimization II
MATH529 Fundamentals of Optimization Unconstrained Optimization II Marco A. Montes de Oca Mathematical Sciences, University of Delaware, USA 1 / 31 Recap 2 / 31 Example Find the local and global minimizers
More informationReal Analysis III. (MAT312β) Department of Mathematics University of Ruhuna. A.W.L. Pubudu Thilan
Real Analysis III (MAT312β) Department of Mathematics University of Ruhuna A.W.L. Pubudu Thilan Department of Mathematics University of Ruhuna Real Analysis III(MAT312β) 1/87 About course unit Course unit:
More informationChapter 7. Extremal Problems. 7.1 Extrema and Local Extrema
Chapter 7 Extremal Problems No matter in theoretical context or in applications many problems can be formulated as problems of finding the maximum or minimum of a function. Whenever this is the case, advanced
More informationMath (P)refresher Lecture 8: Unconstrained Optimization
Math (P)refresher Lecture 8: Unconstrained Optimization September 2006 Today s Topics : Quadratic Forms Definiteness of Quadratic Forms Maxima and Minima in R n First Order Conditions Second Order Conditions
More informationMath General Topology Fall 2012 Homework 1 Solutions
Math 535 - General Topology Fall 2012 Homework 1 Solutions Definition. Let V be a (real or complex) vector space. A norm on V is a function : V R satisfying: 1. Positivity: x 0 for all x V and moreover
More informationChapter 2. Vectors and Vector Spaces
2.1. Operations on Vectors 1 Chapter 2. Vectors and Vector Spaces Section 2.1. Operations on Vectors Note. In this section, we define several arithmetic operations on vectors (especially, vector addition
More informationMATH 304 Linear Algebra Lecture 19: Least squares problems (continued). Norms and inner products.
MATH 304 Linear Algebra Lecture 19: Least squares problems (continued). Norms and inner products. Orthogonal projection Theorem 1 Let V be a subspace of R n. Then any vector x R n is uniquely represented
More informationwhich has a check digit of 9. This is consistent with the first nine digits of the ISBN, since
vector Then the check digit c is computed using the following procedure: 1. Form the dot product. 2. Divide by 11, thereby producing a remainder c that is an integer between 0 and 10, inclusive. The check
More informationg(t) = f(x 1 (t),..., x n (t)).
Reading: [Simon] p. 313-333, 833-836. 0.1 The Chain Rule Partial derivatives describe how a function changes in directions parallel to the coordinate axes. Now we shall demonstrate how the partial derivatives
More informationConvex Optimization and Modeling
Convex Optimization and Modeling Introduction and a quick repetition of analysis/linear algebra First lecture, 12.04.2010 Jun.-Prof. Matthias Hein Organization of the lecture Advanced course, 2+2 hours,
More informationLinear Algebra Massoud Malek
CSUEB Linear Algebra Massoud Malek Inner Product and Normed Space In all that follows, the n n identity matrix is denoted by I n, the n n zero matrix by Z n, and the zero vector by θ n An inner product
More informationAnalysis-3 lecture schemes
Analysis-3 lecture schemes (with Homeworks) 1 Csörgő István November, 2015 1 A jegyzet az ELTE Informatikai Kar 2015. évi Jegyzetpályázatának támogatásával készült Contents 1. Lesson 1 4 1.1. The Space
More informationA LITTLE REAL ANALYSIS AND TOPOLOGY
A LITTLE REAL ANALYSIS AND TOPOLOGY 1. NOTATION Before we begin some notational definitions are useful. (1) Z = {, 3, 2, 1, 0, 1, 2, 3, }is the set of integers. (2) Q = { a b : aεz, bεz {0}} is the set
More informationOn the interior of the simplex, we have the Hessian of d(x), Hd(x) is diagonal with ith. µd(w) + w T c. minimize. subject to w T 1 = 1,
Math 30 Winter 05 Solution to Homework 3. Recognizing the convexity of g(x) := x log x, from Jensen s inequality we get d(x) n x + + x n n log x + + x n n where the equality is attained only at x = (/n,...,
More informationChapter 2: Unconstrained Extrema
Chapter 2: Unconstrained Extrema Math 368 c Copyright 2012, 2013 R Clark Robinson May 22, 2013 Chapter 2: Unconstrained Extrema 1 Types of Sets Definition For p R n and r > 0, the open ball about p of
More informationMath 207 Honors Calculus III Final Exam Solutions
Math 207 Honors Calculus III Final Exam Solutions PART I. Problem 1. A particle moves in the 3-dimensional space so that its velocity v(t) and acceleration a(t) satisfy v(0) = 3j and v(t) a(t) = t 3 for
More informationConsequences of Orthogonality
Consequences of Orthogonality Philippe B. Laval KSU Today Philippe B. Laval (KSU) Consequences of Orthogonality Today 1 / 23 Introduction The three kind of examples we did above involved Dirichlet, Neumann
More informationUNIVERSITY OF NORTH CAROLINA CHARLOTTE 1995 HIGH SCHOOL MATHEMATICS CONTEST March 13, 1995 (C) 10 3 (D) = 1011 (10 1) 9
UNIVERSITY OF NORTH CAROLINA CHARLOTTE 5 HIGH SCHOOL MATHEMATICS CONTEST March, 5. 0 2 0 = (A) (B) 0 (C) 0 (D) 0 (E) 0 (E) 0 2 0 = 0 (0 ) = 0 2. If z = x, what are all the values of y for which (x + y)
More informationCHAPTER 4: HIGHER ORDER DERIVATIVES. Likewise, we may define the higher order derivatives. f(x, y, z) = xy 2 + e zx. y = 2xy.
April 15, 2009 CHAPTER 4: HIGHER ORDER DERIVATIVES In this chapter D denotes an open subset of R n. 1. Introduction Definition 1.1. Given a function f : D R we define the second partial derivatives as
More informationECON 5111 Mathematical Economics
Test 1 October 1, 2010 1. Construct a truth table for the following statement: [p (p q)] q. 2. A prime number is a natural number that is divisible by 1 and itself only. Let P be the set of all prime numbers
More informationj=1 [We will show that the triangle inequality holds for each p-norm in Chapter 3 Section 6.] The 1-norm is A F = tr(a H A).
Math 344 Lecture #19 3.5 Normed Linear Spaces Definition 3.5.1. A seminorm on a vector space V over F is a map : V R that for all x, y V and for all α F satisfies (i) x 0 (positivity), (ii) αx = α x (scale
More informationExercise Sheet 1.
Exercise Sheet 1 You can download my lecture and exercise sheets at the address http://sami.hust.edu.vn/giang-vien/?name=huynt 1) Let A, B be sets. What does the statement "A is not a subset of B " mean?
More informationDefinitions and Properties of R N
Definitions and Properties of R N R N as a set As a set R n is simply the set of all ordered n-tuples (x 1,, x N ), called vectors. We usually denote the vector (x 1,, x N ), (y 1,, y N ), by x, y, or
More information2-7 Solving Quadratic Inequalities. ax 2 + bx + c > 0 (a 0)
Quadratic Inequalities In One Variable LOOKS LIKE a quadratic equation but Doesn t have an equal sign (=) Has an inequality sign (>,
More informationb 1 b 2.. b = b m A = [a 1,a 2,...,a n ] where a 1,j a 2,j a j = a m,j Let A R m n and x 1 x 2 x = x n
Lectures -2: Linear Algebra Background Almost all linear and nonlinear problems in scientific computation require the use of linear algebra These lectures review basic concepts in a way that has proven
More informationALGEBRAIC GEOMETRY HOMEWORK 3
ALGEBRAIC GEOMETRY HOMEWORK 3 (1) Consider the curve Y 2 = X 2 (X + 1). (a) Sketch the curve. (b) Determine the singular point P on C. (c) For all lines through P, determine the intersection multiplicity
More informationMATHEMATICAL ECONOMICS: OPTIMIZATION. Contents
MATHEMATICAL ECONOMICS: OPTIMIZATION JOÃO LOPES DIAS Contents 1. Introduction 2 1.1. Preliminaries 2 1.2. Optimal points and values 2 1.3. The optimization problems 3 1.4. Existence of optimal points 4
More informationLecture 23: 6.1 Inner Products
Lecture 23: 6.1 Inner Products Wei-Ta Chu 2008/12/17 Definition An inner product on a real vector space V is a function that associates a real number u, vwith each pair of vectors u and v in V in such
More informationFALL 2018 MATH 4211/6211 Optimization Homework 1
FALL 2018 MATH 4211/6211 Optimization Homework 1 This homework assignment is open to textbook, reference books, slides, and online resources, excluding any direct solution to the problem (such as solution
More informationMultivariable Calculus
2 Multivariable Calculus 2.1 Limits and Continuity Problem 2.1.1 (Fa94) Let the function f : R n R n satisfy the following two conditions: (i) f (K ) is compact whenever K is a compact subset of R n. (ii)
More informationLinear Algebra. Session 12
Linear Algebra. Session 12 Dr. Marco A Roque Sol 08/01/2017 Example 12.1 Find the constant function that is the least squares fit to the following data x 0 1 2 3 f(x) 1 0 1 2 Solution c = 1 c = 0 f (x)
More informationLINEAR ALGEBRA W W L CHEN
LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008. This chapter is available free to all individuals, on the understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,
More informationMathematics (MEI) Advanced Subsidiary GCE Core 1 (4751) May 2010
Link to past paper on OCR website: http://www.mei.org.uk/files/papers/c110ju_ergh.pdf These solutions are for your personal use only. DO NOT photocopy or pass on to third parties. If you are a school or
More informationIn English, this means that if we travel on a straight line between any two points in C, then we never leave C.
Convex sets In this section, we will be introduced to some of the mathematical fundamentals of convex sets. In order to motivate some of the definitions, we will look at the closest point problem from
More informationIntroduction to Proofs
Real Analysis Preview May 2014 Properties of R n Recall Oftentimes in multivariable calculus, we looked at properties of vectors in R n. If we were given vectors x =< x 1, x 2,, x n > and y =< y1, y 2,,
More informationMath 212-Lecture Interior critical points of functions of two variables
Math 212-Lecture 24 13.10. Interior critical points of functions of two variables Previously, we have concluded that if f has derivatives, all interior local min or local max should be critical points.
More informationEC /11. Math for Microeconomics September Course, Part II Lecture Notes. Course Outline
LONDON SCHOOL OF ECONOMICS Professor Leonardo Felli Department of Economics S.478; x7525 EC400 20010/11 Math for Microeconomics September Course, Part II Lecture Notes Course Outline Lecture 1: Tools for
More informationLecture # 3 Orthogonal Matrices and Matrix Norms. We repeat the definition an orthogonal set and orthornormal set.
Lecture # 3 Orthogonal Matrices and Matrix Norms We repeat the definition an orthogonal set and orthornormal set. Definition A set of k vectors {u, u 2,..., u k }, where each u i R n, is said to be an
More informationBindel, Fall 2011 Intro to Scientific Computing (CS 3220) Week 3: Wednesday, Jan 9
Problem du jour Week 3: Wednesday, Jan 9 1. As a function of matrix dimension, what is the asymptotic complexity of computing a determinant using the Laplace expansion (cofactor expansion) that you probably
More informationThe Conjugate Gradient Method
The Conjugate Gradient Method Jason E. Hicken Aerospace Design Lab Department of Aeronautics & Astronautics Stanford University 14 July 2011 Lecture Objectives describe when CG can be used to solve Ax
More informationMATH Max-min Theory Fall 2016
MATH 20550 Max-min Theory Fall 2016 1. Definitions and main theorems Max-min theory starts with a function f of a vector variable x and a subset D of the domain of f. So far when we have worked with functions
More information11.1 Three-Dimensional Coordinate System
11.1 Three-Dimensional Coordinate System In three dimensions, a point has three coordinates: (x,y,z). The normal orientation of the x, y, and z-axes is shown below. The three axes divide the region into
More informationChapter 13: Vectors and the Geometry of Space
Chapter 13: Vectors and the Geometry of Space 13.1 3-Dimensional Coordinate System 13.2 Vectors 13.3 The Dot Product 13.4 The Cross Product 13.5 Equations of Lines and Planes 13.6 Cylinders and Quadratic
More informationChapter 13: Vectors and the Geometry of Space
Chapter 13: Vectors and the Geometry of Space 13.1 3-Dimensional Coordinate System 13.2 Vectors 13.3 The Dot Product 13.4 The Cross Product 13.5 Equations of Lines and Planes 13.6 Cylinders and Quadratic
More informationMA 102 (Multivariable Calculus)
MA 102 (Multivariable Calculus) Rupam Barman and Shreemayee Bora Department of Mathematics IIT Guwahati Outline of the Course Two Topics: Multivariable Calculus Will be taught as the first part of the
More information1 Lagrange Multiplier Method
1 Lagrange Multiplier Method Near a maximum the decrements on both sides are in the beginning only imperceptible. J. Kepler When a quantity is greatest or least, at that moment its flow neither increases
More informationB553 Lecture 3: Multivariate Calculus and Linear Algebra Review
B553 Lecture 3: Multivariate Calculus and Linear Algebra Review Kris Hauser December 30, 2011 We now move from the univariate setting to the multivariate setting, where we will spend the rest of the class.
More informationMath 4263 Homework Set 1
Homework Set 1 1. Solve the following PDE/BVP 2. Solve the following PDE/BVP 2u t + 3u x = 0 u (x, 0) = sin (x) u x + e x u y = 0 u (0, y) = y 2 3. (a) Find the curves γ : t (x (t), y (t)) such that that
More information3.5 Quadratic Approximation and Convexity/Concavity
3.5 Quadratic Approximation and Convexity/Concavity 55 3.5 Quadratic Approximation and Convexity/Concavity Overview: Second derivatives are useful for understanding how the linear approximation varies
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 informationx 1. x n i + x 2 j (x 1, x 2, x 3 ) = x 1 j + x 3
Version: 4/1/06. Note: These notes are mostly from my 5B course, with the addition of the part on components and projections. Look them over to make sure that we are on the same page as regards inner-products,
More informationChapter 2: Preliminaries and elements of convex analysis
Chapter 2: Preliminaries and elements of convex analysis Edoardo Amaldi DEIB Politecnico di Milano edoardo.amaldi@polimi.it Website: http://home.deib.polimi.it/amaldi/opt-14-15.shtml Academic year 2014-15
More informationFunctional Analysis MATH and MATH M6202
Functional Analysis MATH 36202 and MATH M6202 1 Inner Product Spaces and Normed Spaces Inner Product Spaces Functional analysis involves studying vector spaces where we additionally have the notion of
More informationMathematics (MEI) Advanced Subsidiary GCE Core 1 (4751) June 2010
Link to past paper on OCR website: www.ocr.org.uk The above link takes you to OCR s website. From there you click QUALIFICATIONS, QUALIFICATIONS BY TYPE, AS/A LEVEL GCE, MATHEMATICS (MEI), VIEW ALL DOCUMENTS,
More informationThis pre-publication material is for review purposes only. Any typographical or technical errors will be corrected prior to publication.
This pre-publication material is for review purposes only. Any typographical or technical errors will be corrected prior to publication. Copyright Pearson Canada Inc. All rights reserved. Copyright Pearson
More informationAnother consequence of the Cauchy Schwarz inequality is the continuity of the inner product.
. Inner product spaces 1 Theorem.1 (Cauchy Schwarz inequality). If X is an inner product space then x,y x y. (.) Proof. First note that 0 u v v u = u v u v Re u,v. (.3) Therefore, Re u,v u v (.) for all
More information1 Overview. 2 A Characterization of Convex Functions. 2.1 First-order Taylor approximation. AM 221: Advanced Optimization Spring 2016
AM 221: Advanced Optimization Spring 2016 Prof. Yaron Singer Lecture 8 February 22nd 1 Overview In the previous lecture we saw characterizations of optimality in linear optimization, and we reviewed the
More informationMath Precalculus I University of Hawai i at Mānoa Spring
Math 135 - Precalculus I University of Hawai i at Mānoa Spring - 2014 Created for Math 135, Spring 2008 by Lukasz Grabarek and Michael Joyce Send comments and corrections to lukasz@math.hawaii.edu Contents
More informationLecture 7: Positive Semidefinite Matrices
Lecture 7: Positive Semidefinite Matrices Rajat Mittal IIT Kanpur The main aim of this lecture note is to prepare your background for semidefinite programming. We have already seen some linear algebra.
More informationMath Precalculus I University of Hawai i at Mānoa Spring
Math 135 - Precalculus I University of Hawai i at Mānoa Spring - 2013 Created for Math 135, Spring 2008 by Lukasz Grabarek and Michael Joyce Send comments and corrections to lukasz@math.hawaii.edu Contents
More informationGradient Descent. Dr. Xiaowei Huang
Gradient Descent Dr. Xiaowei Huang https://cgi.csc.liv.ac.uk/~xiaowei/ Up to now, Three machine learning algorithms: decision tree learning k-nn linear regression only optimization objectives are discussed,
More informationLecture 7. Econ August 18
Lecture 7 Econ 2001 2015 August 18 Lecture 7 Outline First, the theorem of the maximum, an amazing result about continuity in optimization problems. Then, we start linear algebra, mostly looking at familiar
More informationFirst Derivative Test
MA 2231 Lecture 22 - Concavity and Relative Extrema Wednesday, November 1, 2017 Objectives: Introduce the Second Derivative Test and its limitations. First Derivative Test When looking for relative extrema
More informationOR MSc Maths Revision Course
OR MSc Maths Revision Course Tom Byrne School of Mathematics University of Edinburgh t.m.byrne@sms.ed.ac.uk 15 September 2017 General Information Today JCMB Lecture Theatre A, 09:30-12:30 Mathematics revision
More informationFunctional Analysis Exercise Class
Functional Analysis Exercise Class Week 9 November 13 November Deadline to hand in the homeworks: your exercise class on week 16 November 20 November Exercises (1) Show that if T B(X, Y ) and S B(Y, Z)
More informationStructural and Multidisciplinary Optimization. P. Duysinx and P. Tossings
Structural and Multidisciplinary Optimization P. Duysinx and P. Tossings 2018-2019 CONTACTS Pierre Duysinx Institut de Mécanique et du Génie Civil (B52/3) Phone number: 04/366.91.94 Email: P.Duysinx@uliege.be
More information1. Subspaces A subset M of Hilbert space H is a subspace of it is closed under the operation of forming linear combinations;i.e.,
Abstract Hilbert Space Results We have learned a little about the Hilbert spaces L U and and we have at least defined H 1 U and the scale of Hilbert spaces H p U. Now we are going to develop additional
More informationProjection Theorem 1
Projection Theorem 1 Cauchy-Schwarz Inequality Lemma. (Cauchy-Schwarz Inequality) For all x, y in an inner product space, [ xy, ] x y. Equality holds if and only if x y or y θ. Proof. If y θ, the inequality
More informationPolynomial Functions
Polynomial Functions NOTE: Some problems in this file are used with permission from the engageny.org website of the New York State Department of Education. Various files. Internet. Available from https://www.engageny.org/ccss-library.
More informationDesigning Information Devices and Systems I Fall 2018 Lecture Notes Note 21
EECS 16A Designing Information Devices and Systems I Fall 2018 Lecture Notes Note 21 21.1 Module Goals In this module, we introduce a family of ideas that are connected to optimization and machine learning,
More information100 CHAPTER 4. SYSTEMS AND ADAPTIVE STEP SIZE METHODS APPENDIX
100 CHAPTER 4. SYSTEMS AND ADAPTIVE STEP SIZE METHODS APPENDIX.1 Norms If we have an approximate solution at a given point and we want to calculate the absolute error, then we simply take the magnitude
More informationa. Define a function called an inner product on pairs of points x = (x 1, x 2,..., x n ) and y = (y 1, y 2,..., y n ) in R n by
Real Analysis Homework 1 Solutions 1. Show that R n with the usual euclidean distance is a metric space. Items a-c will guide you through the proof. a. Define a function called an inner product on pairs
More informationExtra Problems for Math 2050 Linear Algebra I
Extra Problems for Math 5 Linear Algebra I Find the vector AB and illustrate with a picture if A = (,) and B = (,4) Find B, given A = (,4) and [ AB = A = (,4) and [ AB = 8 If possible, express x = 7 as
More informationMath Linear Algebra II. 1. Inner Products and Norms
Math 342 - Linear Algebra II Notes 1. Inner Products and Norms One knows from a basic introduction to vectors in R n Math 254 at OSU) that the length of a vector x = x 1 x 2... x n ) T R n, denoted x,
More informationwhere the bar indicates complex conjugation. Note that this implies that, from Property 2, x,αy = α x,y, x,y X, α C.
Lecture 4 Inner product spaces Of course, you are familiar with the idea of inner product spaces at least finite-dimensional ones. Let X be an abstract vector space with an inner product, denoted as,,
More informationDS-GA 1002 Lecture notes 0 Fall Linear Algebra. These notes provide a review of basic concepts in linear algebra.
DS-GA 1002 Lecture notes 0 Fall 2016 Linear Algebra These notes provide a review of basic concepts in linear algebra. 1 Vector spaces You are no doubt familiar with vectors in R 2 or R 3, i.e. [ ] 1.1
More informationOptimization and Optimal Control in Banach Spaces
Optimization and Optimal Control in Banach Spaces Bernhard Schmitzer October 19, 2017 1 Convex non-smooth optimization with proximal operators Remark 1.1 (Motivation). Convex optimization: easier to solve,
More information1. Fill in the three character code you received via in the box
September 18, 2001 Your name The first 20 problems count 3 points each and the final ones counts as marked. Problems 1 through 20 are multiple choice. In the multiple choice section, circle the correct
More informationLinear Algebra. Chapter 8: Eigenvalues: Further Applications and Computations Section 8.2. Applications to Geometry Proofs of Theorems.
Linear Algebra Chapter 8: Eigenvalues: Further Applications and Computations Section 8.2. Applications to Geometry Proofs of Theorems May 1, 2018 () Linear Algebra May 1, 2018 1 / 8 Table of contents 1
More informationConvergence of sequences, limit of functions, continuity
Convergence of sequences, limit of functions, continuity With the definition of norm, or more precisely the distance between any two vectors in R N : dist(x, y) 7 x y 7 [(x 1 y 1 ) 2 + + (x N y N ) 2 ]
More informationINNER PRODUCT SPACE. Definition 1
INNER PRODUCT SPACE Definition 1 Suppose u, v and w are all vectors in vector space V and c is any scalar. An inner product space on the vectors space V is a function that associates with each pair of
More information2x (x 2 + y 2 + 1) 2 2y. (x 2 + y 2 + 1) 4. 4xy. (1, 1)(x 1) + (1, 1)(y + 1) (1, 1)(x 1)(y + 1) 81 x y y + 7.
Homework 8 Solutions, November 007. (1 We calculate some derivatives: f x = f y = x (x + y + 1 y (x + y + 1 x = (x + y + 1 4x (x + y + 1 4 y = (x + y + 1 4y (x + y + 1 4 x y = 4xy (x + y + 1 4 Substituting
More informationMatrix Algebra: Vectors
A Matrix Algebra: Vectors A Appendix A: MATRIX ALGEBRA: VECTORS A 2 A MOTIVATION Matrix notation was invented primarily to express linear algebra relations in compact form Compactness enhances visualization
More informationSeveral variables. x 1 x 2. x n
Several variables Often we have not only one, but several variables in a problem The issues that come up are somewhat more complex than for one variable Let us first start with vector spaces and linear
More informationThe Derivative. Appendix B. B.1 The Derivative of f. Mappings from IR to IR
Appendix B The Derivative B.1 The Derivative of f In this chapter, we give a short summary of the derivative. Specifically, we want to compare/contrast how the derivative appears for functions whose domain
More informationDetailed objectives are given in each of the sections listed below. 1. Cartesian Space Coordinates. 2. Displacements, Forces, Velocities and Vectors
Unit 1 Vectors In this unit, we introduce vectors, vector operations, and equations of lines and planes. Note: Unit 1 is based on Chapter 12 of the textbook, Salas and Hille s Calculus: Several Variables,
More informationMath 110: Worksheet 1 Solutions
Math 110: Worksheet 1 Solutions August 30 Thursday Aug. 4 1. Determine whether or not the following sets form vector spaces over the given fields. (a) The set V of all matrices of the form where a, b R,
More informationA linear equation in two variables is generally written as follows equation in three variables can be written as
System of Equations A system of equations is a set of equations considered simultaneously. In this course, we will discuss systems of equation in two or three variables either linear or quadratic or a
More informationMax-Min Problems in R n Matrix
Max-Min Problems in R n Matrix 1 and the essian Prerequisite: Section 6., Orthogonal Diagonalization n this section, we study the problem of nding local maxima and minima for realvalued functions on R
More informationPart 1a: Inner product, Orthogonality, Vector/Matrix norm
Part 1a: Inner product, Orthogonality, Vector/Matrix norm September 19, 2018 Numerical Linear Algebra Part 1a September 19, 2018 1 / 16 1. Inner product on a linear space V over the number field F A map,
More informationMath 273a: Optimization Basic concepts
Math 273a: Optimization Basic concepts Instructor: Wotao Yin Department of Mathematics, UCLA Spring 2015 slides based on Chong-Zak, 4th Ed. Goals of this lecture The general form of optimization: minimize
More informationGrothendieck s Inequality
Grothendieck s Inequality Leqi Zhu 1 Introduction Let A = (A ij ) R m n be an m n matrix. Then A defines a linear operator between normed spaces (R m, p ) and (R n, q ), for 1 p, q. The (p q)-norm of A
More informationMath 10C - Fall Final Exam
Math 1C - Fall 217 - Final Exam Problem 1. Consider the function f(x, y) = 1 x 2 (y 1) 2. (i) Draw the level curve through the point P (1, 2). Find the gradient of f at the point P and draw the gradient
More informationREVIEW OF DIFFERENTIAL CALCULUS
REVIEW OF DIFFERENTIAL CALCULUS DONU ARAPURA 1. Limits and continuity To simplify the statements, we will often stick to two variables, but everything holds with any number of variables. Let f(x, y) be
More informationWorkshop I The R n Vector Space. Linear Combinations
Workshop I Workshop I The R n Vector Space. Linear Combinations Exercise 1. Given the vectors u = (1, 0, 3), v = ( 2, 1, 2), and w = (1, 2, 4) compute a) u + ( v + w) b) 2 u 3 v c) 2 v u + w Exercise 2.
More informationHOMEWORK PROBLEMS FROM STRANG S LINEAR ALGEBRA AND ITS APPLICATIONS (4TH EDITION)
HOMEWORK PROBLEMS FROM STRANG S LINEAR ALGEBRA AND ITS APPLICATIONS (4TH EDITION) PROFESSOR STEVEN MILLER: BROWN UNIVERSITY: SPRING 2007 1. CHAPTER 1: MATRICES AND GAUSSIAN ELIMINATION Page 9, # 3: Describe
More informationPre-Algebra 2. Unit 9. Polynomials Name Period
Pre-Algebra Unit 9 Polynomials Name Period 9.1A Add, Subtract, and Multiplying Polynomials (non-complex) Explain Add the following polynomials: 1) ( ) ( ) ) ( ) ( ) Subtract the following polynomials:
More informationSection 4.2. Types of Differentiation
42 Types of Differentiation 1 Section 42 Types of Differentiation Note In this section we define differentiation of various structures with respect to a scalar, a vector, and a matrix Definition Let vector
More informationorthogonal relations between vectors and subspaces Then we study some applications in vector spaces and linear systems, including Orthonormal Basis,
5 Orthogonality Goals: We use scalar products to find the length of a vector, the angle between 2 vectors, projections, orthogonal relations between vectors and subspaces Then we study some applications
More information