Orthogonal Projections and Orthonormal Bases


 Sarah Nicholson
 6 years ago
 Views:
Transcription
1 CS 3, HANDOUT A, 3 November 04 (adjusted on 7 November 04) Orthogonal Projections and Orthonormal Bases (continuation of Handout 07 of 6 September 04)
2 Definition (Orthogonality, length, unit vectors). Two vectors v, w R n are orthogonal if v w = 0.. The length (or norm) of a vector v R n is v = v v. 3. A vector u R n is a unit vector if u =. Definition (Orthonormal vectors) The vectors u,..., u k R n are called orthonormal if:. u = = u k =,. u i u j = 0 for all i < j k.
3 Theorem 3 (Properties of orthonormal vectors). Orthonormal vectors are linearly independent.. If the vectors u,..., u n R n are orthonormal, they form a basis of R n. Theorem 4 (Orthogonal projection) Consider a vector x R n and a subspace V of R n. Then we can write x = x + x where x is in V and x is perpendicular to V, and this representation is unique. x is called the orthogonal projection of x on V. 3
4 Example of orthonormal vectors in the hyperspace R 4 The vectors: u =, u =, u 3 = are orthonormal. (Why?). Is Span{ u, u, u 3 } = R 4? No. (Why?). Are there unit vectors u 4 in R 4 that are orthogonal to u, u, and u 3? Yes, there are exactly two such unit vectors u 4. (Why?) 3. For such a vector R 4 as in part, if it exists, is it the case that Span{ u, u, u 3,, u 4 } = R 4? Yes. (Why?) 4
5 Some intuition from R and R 3, helpful in understanding the dot product (see pp in [Lay]): If u and v are nonzero vectors in R or R 3, then there is a nice connection between the dot product u v and the angle θ between the two vectors. As usual, we assume one common endpoint of u and v is the origin, and their respective second endpoint is determined by two coordinates (in R ) or three coordinates (in R 3 ). We verify this formula for: u v = u v cos θ θ = 0, in which case cos θ = and u v = u v. θ = 90, in which case cos θ = 0 and u v = 0. θ in general, see pages in [Lay]. This verification is not possible in hyperspace R n, with n 4, but is still helpful in understanding the next theorem. 5
6 Theorem 5 (Formula for orthogonal projection) [Theorem 8, page 348, and Theorem 0, page 35, in [Lay]]. Let V be a subspace of R n with orthogonal basis u,..., u k R n. Then, for every x R n, we have proj V ( x) = x = u x u u u + + u k x u k u k u k. Let V be a subspace of R n with orthonormal basis u,..., u k R n. Then, for every x R n, we have proj V ( x) = x = ( u x) u + + ( u k x) u k Corollary 6 Let V = R n with orthonormal basis u,..., u n R n. Then, for every x R n, we have x = ( u x) u + + ( u n x) u n 6
7 The following is a simpler restatement of the preceding corollary: Corollary 7 If u,..., u n R n is an orthonormal basis for R n, then any vector x R n can be uniquely expressed as a linear combination of the basis: x = c u + + c n u n where c i = u i x for every i n. 7
8 Definition 8 (Orthogonal complement) Let V be a subspace of R n. The orthogonal complement V of V is the set of all vectors x in R n that are orthogonal to all vectors in V : V = { x R n v x = 0 for every v V } Remark 9 If V is a subspace of R n, then the orthogonal projection proj V ( ) on V is a linear transformation from R n to V. (You believe this?) Then the kernel of the orthogonal projection, i.e., ker(proj V ( )), is precisely the orthogonal complement V. Theorem 0 (Properties of the orthogonal complement) Let V be a subspace of R n.. The orthogonal complement V if V is a subspace of R n.. The intersection of V and V consists of the zero vector: V V = O. 3. dim(v ) + dim(v ) = n. 8
9 Exercise : Let V be a subspace of R n with orthonormal basis u,..., u k R n. Show that the orthogonal projection proj V ( ) on V is a linear transformation from R n to V. Hint: Show that the two defining properties of a linear transformation (i) and (ii) on page 65 of [Lay] are satisfied by proj V ( ). Exercise : Let V be the subspace V of R 4 spanned by two vectors u = (,,, ) and u = (,,, ).. Find an orthonormal basis for V.. Find proj V ( x) where x = (, 3,, 7) and verify that ( x proj V ( x)) is perpendicular to both u and u. 3. By the preceding exercise, proj V ( ) is a linear transformation. Find a 4 4 matrix A that represents the linear transformation proj V ( ). Hint for part : This is not trivial. You may get some help by searching the Web with the key Find the matrix of a linear transformation or some such. It is helpful to keep in mind that the particular V in this exercise has dimension as a subspace of R 4, so that rank(a) = dim(im(a)) = dim(v ) = and dim(ker(a)) = dim(v ) =. 9
10 Theorem 3 (Pythagorean theorem) [Theorem, page 334, [Lay] ] Let x, y R n. The equation x + y = x + y holds if and only if x and y are orthogonal. Theorem 4 (Inequality for the magnitude of proj V ( x)) Let V be a subspace of R n and x R n. Then proj V ( x) x The statement is an equality iff x V. 0
α = u v. In other words, Orthogonal Projection
Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v
More informationLecture 14: Section 3.3
Lecture 14: Section 3.3 Shuanglin Shao October 23, 2013 Definition. Two nonzero vectors u and v in R n are said to be orthogonal (or perpendicular) if u v = 0. We will also agree that the zero vector in
More informationSection 5.3. Section 5.3. u m ] l jj. = l jj u j + + l mj u m. v j = [ u 1 u j. l mj
Section 5. l j v j = [ u u j u m ] l jj = l jj u j + + l mj u m. l mj Section 5. 5.. Not orthogonal, the column vectors fail to be perpendicular to each other. 5..2 his matrix is orthogonal. Check that
More informationMAT 1341: REVIEW II SANGHOON BAEK
MAT 1341: REVIEW II SANGHOON BAEK 1. Projections and Cross Product 1.1. Projections. Definition 1.1. Given a vector u, the rectangular (or perpendicular or orthogonal) components are two vectors u 1 and
More informationMath 215 HW #6 Solutions
Math 5 HW #6 Solutions Problem 34 Show that x y is orthogonal to x + y if and only if x = y Proof First, suppose x y is orthogonal to x + y Then since x, y = y, x In other words, = x y, x + y = (x y) T
More information1.3. DOT PRODUCT 19. 6. If θ is the angle (between 0 and π) between two nonzero vectors u and v,
1.3. DOT PRODUCT 19 1.3 Dot Product 1.3.1 Definitions and Properties The dot product is the first way to multiply two vectors. The definition we will give below may appear arbitrary. But it is not. It
More informationLectures notes on orthogonal matrices (with exercises) 92.222  Linear Algebra II  Spring 2004 by D. Klain
Lectures notes on orthogonal matrices (with exercises) 92.222  Linear Algebra II  Spring 2004 by D. Klain 1. Orthogonal matrices and orthonormal sets An n n realvalued matrix A is said to be an orthogonal
More informationDot product and vector projections (Sect. 12.3) There are two main ways to introduce the dot product
Dot product and vector projections (Sect. 12.3) Two definitions for the dot product. Geometric definition of dot product. Orthogonal vectors. Dot product and orthogonal projections. Properties of the dot
More information1 VECTOR SPACES AND SUBSPACES
1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such
More informationAdding vectors We can do arithmetic with vectors. We ll start with vector addition and related operations. Suppose you have two vectors
1 Chapter 13. VECTORS IN THREE DIMENSIONAL SPACE Let s begin with some names and notation for things: R is the set (collection) of real numbers. We write x R to mean that x is a real number. A real number
More informationRecall that two vectors in are perpendicular or orthogonal provided that their dot
Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal
More informationEquations Involving Lines and Planes Standard equations for lines in space
Equations Involving Lines and Planes In this section we will collect various important formulas regarding equations of lines and planes in three dimensional space Reminder regarding notation: any quantity
More informationA vector is a directed line segment used to represent a vector quantity.
Chapters and 6 Introduction to Vectors A vector quantity has direction and magnitude. There are many examples of vector quantities in the natural world, such as force, velocity, and acceleration. A vector
More informationi=(1,0), j=(0,1) in R 2 i=(1,0,0), j=(0,1,0), k=(0,0,1) in R 3 e 1 =(1,0,..,0), e 2 =(0,1,,0),,e n =(0,0,,1) in R n.
Length, norm, magnitude of a vector v=(v 1,,v n ) is v = (v 12 +v 22 + +v n2 ) 1/2. Examples v=(1,1,,1) v =n 1/2. Unit vectors u=v/ v corresponds to directions. Standard unit vectors i=(1,0), j=(0,1) in
More information3. INNER PRODUCT SPACES
. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.
More informationChapter 6. Orthogonality
6.3 Orthogonal Matrices 1 Chapter 6. Orthogonality 6.3 Orthogonal Matrices Definition 6.4. An n n matrix A is orthogonal if A T A = I. Note. We will see that the columns of an orthogonal matrix must be
More informationCross product and determinants (Sect. 12.4) Two main ways to introduce the cross product
Cross product and determinants (Sect. 12.4) Two main ways to introduce the cross product Geometrical definition Properties Expression in components. Definition in components Properties Geometrical expression.
More information9 Multiplication of Vectors: The Scalar or Dot Product
Arkansas Tech University MATH 934: Calculus III Dr. Marcel B Finan 9 Multiplication of Vectors: The Scalar or Dot Product Up to this point we have defined what vectors are and discussed basic notation
More informationRecall the basic property of the transpose (for any A): v A t Aw = v w, v, w R n.
ORTHOGONAL MATRICES Informally, an orthogonal n n matrix is the ndimensional analogue of the rotation matrices R θ in R 2. When does a linear transformation of R 3 (or R n ) deserve to be called a rotation?
More information( ) which must be a vector
MATH 37 Linear Transformations from Rn to Rm Dr. Neal, WKU Let T : R n R m be a function which maps vectors from R n to R m. Then T is called a linear transformation if the following two properties are
More informationInner Product Spaces and Orthogonality
Inner Product Spaces and Orthogonality week 34 Fall 2006 Dot product of R n The inner product or dot product of R n is a function, defined by u, v a b + a 2 b 2 + + a n b n for u a, a 2,, a n T, v b,
More informationMath 241, Exam 1 Information.
Math 241, Exam 1 Information. 9/24/12, LC 310, 11:1512:05. Exam 1 will be based on: Sections 12.112.5, 14.114.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)
More informationOrthogonal Diagonalization of Symmetric Matrices
MATH10212 Linear Algebra Brief lecture notes 57 Gram Schmidt Process enables us to find an orthogonal basis of a subspace. Let u 1,..., u k be a basis of a subspace V of R n. We begin the process of finding
More informationMATH1231 Algebra, 2015 Chapter 7: Linear maps
MATH1231 Algebra, 2015 Chapter 7: Linear maps A/Prof. Daniel Chan School of Mathematics and Statistics University of New South Wales danielc@unsw.edu.au Daniel Chan (UNSW) MATH1231 Algebra 1 / 43 Chapter
More informationMethods for Finding Bases
Methods for Finding Bases Bases for the subspaces of a matrix Rowreduction methods can be used to find bases. Let us now look at an example illustrating how to obtain bases for the row space, null space,
More informationSimilarity and Diagonalization. Similar Matrices
MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that
More informationSection 6.1  Inner Products and Norms
Section 6.1  Inner Products and Norms Definition. Let V be a vector space over F {R, C}. An inner product on V is a function that assigns, to every ordered pair of vectors x and y in V, a scalar in F,
More informationx + y + z = 1 2x + 3y + 4z = 0 5x + 6y + 7z = 3
Math 24 FINAL EXAM (2/9/9  SOLUTIONS ( Find the general solution to the system of equations 2 4 5 6 7 ( r 2 2r r 2 r 5r r x + y + z 2x + y + 4z 5x + 6y + 7z 2 2 2 2 So x z + y 2z 2 and z is free. ( r
More informationSection 4.4 Inner Product Spaces
Section 4.4 Inner Product Spaces In our discussion of vector spaces the specific nature of F as a field, other than the fact that it is a field, has played virtually no role. In this section we no longer
More informationChapter 17. Orthogonal Matrices and Symmetries of Space
Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length
More informationOrthogonal Projections
Orthogonal Projections and Reflections (with exercises) by D. Klain Version.. Corrections and comments are welcome! Orthogonal Projections Let X,..., X k be a family of linearly independent (column) vectors
More informationCS3220 Lecture Notes: QR factorization and orthogonal transformations
CS3220 Lecture Notes: QR factorization and orthogonal transformations Steve Marschner Cornell University 11 March 2009 In this lecture I ll talk about orthogonal matrices and their properties, discuss
More informationSection 1.1. Introduction to R n
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to
More informationNumerical Analysis Lecture Notes
Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number
More informationMAT 200, Midterm Exam Solution. a. (5 points) Compute the determinant of the matrix A =
MAT 200, Midterm Exam Solution. (0 points total) a. (5 points) Compute the determinant of the matrix 2 2 0 A = 0 3 0 3 0 Answer: det A = 3. The most efficient way is to develop the determinant along the
More informationISOMETRIES OF R n KEITH CONRAD
ISOMETRIES OF R n KEITH CONRAD 1. Introduction An isometry of R n is a function h: R n R n that preserves the distance between vectors: h(v) h(w) = v w for all v and w in R n, where (x 1,..., x n ) = x
More informationx1 x 2 x 3 y 1 y 2 y 3 x 1 y 2 x 2 y 1 0.
Cross product 1 Chapter 7 Cross product We are getting ready to study integration in several variables. Until now we have been doing only differential calculus. One outcome of this study will be our ability
More informationTHREE DIMENSIONAL GEOMETRY
Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,
More information28 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE. v x. u y v z u z v y u y u z. v y v z
28 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.4 Cross Product 1.4.1 Definitions The cross product is the second multiplication operation between vectors we will study. The goal behind the definition
More informationMAT 242 Test 3 SOLUTIONS, FORM A
MAT Test SOLUTIONS, FORM A. Let v =, v =, and v =. Note that B = { v, v, v } is an orthogonal set. Also, let W be the subspace spanned by { v, v, v }. A = 8 a. [5 points] Find the orthogonal projection
More informationProblem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday.
Math 312, Fall 2012 Jerry L. Kazdan Problem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday. In addition to the problems below, you should also know how to solve
More informationCITY UNIVERSITY LONDON. BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION
No: CITY UNIVERSITY LONDON BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION ENGINEERING MATHEMATICS 2 (resit) EX2005 Date: August
More informationInner Product Spaces
Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and
More informationApplied Linear Algebra I Review page 1
Applied Linear Algebra Review 1 I. Determinants A. Definition of a determinant 1. Using sum a. Permutations i. Sign of a permutation ii. Cycle 2. Uniqueness of the determinant function in terms of properties
More informationMATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets.
MATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets. Norm The notion of norm generalizes the notion of length of a vector in R n. Definition. Let V be a vector space. A function α
More information18.06 Problem Set 4 Solution Due Wednesday, 11 March 2009 at 4 pm in 2106. Total: 175 points.
806 Problem Set 4 Solution Due Wednesday, March 2009 at 4 pm in 206 Total: 75 points Problem : A is an m n matrix of rank r Suppose there are righthandsides b for which A x = b has no solution (a) What
More information[1] Diagonal factorization
8.03 LA.6: Diagonalization and Orthogonal Matrices [ Diagonal factorization [2 Solving systems of first order differential equations [3 Symmetric and Orthonormal Matrices [ Diagonal factorization Recall:
More informationInner product. Definition of inner product
Math 20F Linear Algebra Lecture 25 1 Inner product Review: Definition of inner product. Slide 1 Norm and distance. Orthogonal vectors. Orthogonal complement. Orthogonal basis. Definition of inner product
More information160 CHAPTER 4. VECTOR SPACES
160 CHAPTER 4. VECTOR SPACES 4. Rank and Nullity In this section, we look at relationships between the row space, column space, null space of a matrix and its transpose. We will derive fundamental results
More informationVector Algebra CHAPTER 13. Ü13.1. Basic Concepts
CHAPTER 13 ector Algebra Ü13.1. Basic Concepts A vector in the plane or in space is an arrow: it is determined by its length, denoted and its direction. Two arrows represent the same vector if they have
More informationApplied Linear Algebra
Applied Linear Algebra OTTO BRETSCHER http://www.prenhall.com/bretscher Chapter 7 Eigenvalues and Eigenvectors ChiaHui Chang Email: chia@csie.ncu.edu.tw National Central University, Taiwan 7.1 DYNAMICAL
More informationLinear Algebra Review. Vectors
Linear Algebra Review By Tim K. Marks UCSD Borrows heavily from: Jana Kosecka kosecka@cs.gmu.edu http://cs.gmu.edu/~kosecka/cs682.html Virginia de Sa Cogsci 8F Linear Algebra review UCSD Vectors The length
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More information5.3 The Cross Product in R 3
53 The Cross Product in R 3 Definition 531 Let u = [u 1, u 2, u 3 ] and v = [v 1, v 2, v 3 ] Then the vector given by [u 2 v 3 u 3 v 2, u 3 v 1 u 1 v 3, u 1 v 2 u 2 v 1 ] is called the cross product (or
More informationSolutions to old Exam 1 problems
Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections
More informationThe Dot and Cross Products
The Dot and Cross Products Two common operations involving vectors are the dot product and the cross product. Let two vectors =,, and =,, be given. The Dot Product The dot product of and is written and
More informationis in plane V. However, it may be more convenient to introduce a plane coordinate system in V.
.4 COORDINATES EXAMPLE Let V be the plane in R with equation x +2x 2 +x 0, a twodimensional subspace of R. We can describe a vector in this plane by its spatial (D)coordinates; for example, vector x 5
More informationComputing Orthonormal Sets in 2D, 3D, and 4D
Computing Orthonormal Sets in 2D, 3D, and 4D David Eberly Geometric Tools, LLC http://www.geometrictools.com/ Copyright c 19982016. All Rights Reserved. Created: March 22, 2010 Last Modified: August 11,
More informationThese axioms must hold for all vectors ū, v, and w in V and all scalars c and d.
DEFINITION: A vector space is a nonempty set V of objects, called vectors, on which are defined two operations, called addition and multiplication by scalars (real numbers), subject to the following axioms
More informationLecture Notes 2: Matrices as Systems of Linear Equations
2: Matrices as Systems of Linear Equations 33A Linear Algebra, Puck Rombach Last updated: April 13, 2016 Systems of Linear Equations Systems of linear equations can represent many things You have probably
More informationWHEN DOES A CROSS PRODUCT ON R n EXIST?
WHEN DOES A CROSS PRODUCT ON R n EXIST? PETER F. MCLOUGHLIN It is probably safe to say that just about everyone reading this article is familiar with the cross product and the dot product. However, what
More informationVector Math Computer Graphics Scott D. Anderson
Vector Math Computer Graphics Scott D. Anderson 1 Dot Product The notation v w means the dot product or scalar product or inner product of two vectors, v and w. In abstract mathematics, we can talk about
More informationLinear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices
MATH 30 Differential Equations Spring 006 Linear algebra and the geometry of quadratic equations Similarity transformations and orthogonal matrices First, some things to recall from linear algebra Two
More informationSolutions to Math 51 First Exam January 29, 2015
Solutions to Math 5 First Exam January 29, 25. ( points) (a) Complete the following sentence: A set of vectors {v,..., v k } is defined to be linearly dependent if (2 points) there exist c,... c k R, not
More informationSouth Carolina College and CareerReady (SCCCR) PreCalculus
South Carolina College and CareerReady (SCCCR) PreCalculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know
More informationMath 241 Lines and Planes (Solutions) x = 3 3t. z = 1 t. x = 5 + t. z = 7 + 3t
Math 241 Lines and Planes (Solutions) The equations for planes P 1, P 2 and P are P 1 : x 2y + z = 7 P 2 : x 4y + 5z = 6 P : (x 5) 2(y 6) + (z 7) = 0 The equations for lines L 1, L 2, L, L 4 and L 5 are
More informationBANACH AND HILBERT SPACE REVIEW
BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but
More informationMATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors. Jordan canonical form (continued).
MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors Jordan canonical form (continued) Jordan canonical form A Jordan block is a square matrix of the form λ 1 0 0 0 0 λ 1 0 0 0 0 λ 0 0 J = 0
More informationLinear Algebra: Vectors
A Linear Algebra: Vectors A Appendix A: LINEAR ALGEBRA: VECTORS TABLE OF CONTENTS Page A Motivation A 3 A2 Vectors A 3 A2 Notational Conventions A 4 A22 Visualization A 5 A23 Special Vectors A 5 A3 Vector
More informationa.) Write the line 2x  4y = 9 into slope intercept form b.) Find the slope of the line parallel to part a
Bellwork a.) Write the line 2x  4y = 9 into slope intercept form b.) Find the slope of the line parallel to part a c.) Find the slope of the line perpendicular to part b or a May 8 7:30 AM 1 Day 1 I.
More informationLeastSquares Intersection of Lines
LeastSquares Intersection of Lines Johannes Traa  UIUC 2013 This writeup derives the leastsquares solution for the intersection of lines. In the general case, a set of lines will not intersect at a
More informationNotes from February 11
Notes from February 11 Math 130 Course web site: www.courses.fas.harvard.edu/5811 Two lemmas Before proving the theorem which was stated at the end of class on February 8, we begin with two lemmas. The
More informationNotes on Orthogonal and Symmetric Matrices MENU, Winter 2013
Notes on Orthogonal and Symmetric Matrices MENU, Winter 201 These notes summarize the main properties and uses of orthogonal and symmetric matrices. We covered quite a bit of material regarding these topics,
More informationVector has a magnitude and a direction. Scalar has a magnitude
Vector has a magnitude and a direction Scalar has a magnitude Vector has a magnitude and a direction Scalar has a magnitude a brick on a table Vector has a magnitude and a direction Scalar has a magnitude
More informationSection 1.4. Lines, Planes, and Hyperplanes. The Calculus of Functions of Several Variables
The Calculus of Functions of Several Variables Section 1.4 Lines, Planes, Hyperplanes In this section we will add to our basic geometric understing of R n by studying lines planes. If we do this carefully,
More informationMatrix Representations of Linear Transformations and Changes of Coordinates
Matrix Representations of Linear Transformations and Changes of Coordinates 01 Subspaces and Bases 011 Definitions A subspace V of R n is a subset of R n that contains the zero element and is closed under
More informationGeometry of Vectors. 1 Cartesian Coordinates. Carlo Tomasi
Geometry of Vectors Carlo Tomasi This note explores the geometric meaning of norm, inner product, orthogonality, and projection for vectors. For vectors in threedimensional space, we also examine the
More informationSection 9.5: Equations of Lines and Planes
Lines in 3D Space Section 9.5: Equations of Lines and Planes Practice HW from Stewart Textbook (not to hand in) p. 673 # 35 odd, 237 odd, 4, 47 Consider the line L through the point P = ( x, y, ) that
More information11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space
11 Vectors and the Geometry of Space 11.1 Vectors in the Plane Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. 2 Objectives! Write the component form of
More informationName Class. Date Section. Test Form A Chapter 11. Chapter 11 Test Bank 155
Chapter Test Bank 55 Test Form A Chapter Name Class Date Section. Find a unit vector in the direction of v if v is the vector from P,, 3 to Q,, 0. (a) 3i 3j 3k (b) i j k 3 i 3 j 3 k 3 i 3 j 3 k. Calculate
More informationMATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix.
MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix. Nullspace Let A = (a ij ) be an m n matrix. Definition. The nullspace of the matrix A, denoted N(A), is the set of all ndimensional column
More informationA QUICK GUIDE TO THE FORMULAS OF MULTIVARIABLE CALCULUS
A QUIK GUIDE TO THE FOMULAS OF MULTIVAIABLE ALULUS ontents 1. Analytic Geometry 2 1.1. Definition of a Vector 2 1.2. Scalar Product 2 1.3. Properties of the Scalar Product 2 1.4. Length and Unit Vectors
More informationExact Values of the Sine and Cosine Functions in Increments of 3 degrees
Exact Values of the Sine and Cosine Functions in Increments of 3 degrees The sine and cosine values for all angle measurements in multiples of 3 degrees can be determined exactly, represented in terms
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 understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,
More informationModule 8 Lesson 4: Applications of Vectors
Module 8 Lesson 4: Applications of Vectors So now that you have learned the basic skills necessary to understand and operate with vectors, in this lesson, we will look at how to solve real world problems
More informationNOTES ON LINEAR TRANSFORMATIONS
NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all
More informationLINEAR ALGEBRA. September 23, 2010
LINEAR ALGEBRA September 3, 00 Contents 0. LUdecomposition.................................... 0. Inverses and Transposes................................. 0.3 Column Spaces and NullSpaces.............................
More informationT ( a i x i ) = a i T (x i ).
Chapter 2 Defn 1. (p. 65) Let V and W be vector spaces (over F ). We call a function T : V W a linear transformation form V to W if, for all x, y V and c F, we have (a) T (x + y) = T (x) + T (y) and (b)
More informationLinearly Independent Sets and Linearly Dependent Sets
These notes closely follow the presentation of the material given in David C. Lay s textbook Linear Algebra and its Applications (3rd edition). These notes are intended primarily for inclass presentation
More information13 MATH FACTS 101. 2 a = 1. 7. The elements of a vector have a graphical interpretation, which is particularly easy to see in two or three dimensions.
3 MATH FACTS 0 3 MATH FACTS 3. Vectors 3.. Definition We use the overhead arrow to denote a column vector, i.e., a linear segment with a direction. For example, in threespace, we write a vector in terms
More informationv w is orthogonal to both v and w. the three vectors v, w and v w form a righthanded set of vectors.
3. Cross product Definition 3.1. Let v and w be two vectors in R 3. The cross product of v and w, denoted v w, is the vector defined as follows: the length of v w is the area of the parallelogram with
More informationLinear Algebra Notes
Linear Algebra Notes Chapter 19 KERNEL AND IMAGE OF A MATRIX Take an n m matrix a 11 a 12 a 1m a 21 a 22 a 2m a n1 a n2 a nm and think of it as a function A : R m R n The kernel of A is defined as Note
More information1 0 5 3 3 A = 0 0 0 1 3 0 0 0 0 0 0 0 0 0 0
Solutions: Assignment 4.. Find the redundant column vectors of the given matrix A by inspection. Then find a basis of the image of A and a basis of the kernel of A. 5 A The second and third columns are
More informationLINES AND PLANES CHRIS JOHNSON
LINES AND PLANES CHRIS JOHNSON Abstract. In this lecture we derive the equations for lines and planes living in 3space, as well as define the angle between two nonparallel planes, and determine the distance
More information4.5 Linear Dependence and Linear Independence
4.5 Linear Dependence and Linear Independence 267 32. {v 1, v 2 }, where v 1, v 2 are collinear vectors in R 3. 33. Prove that if S and S are subsets of a vector space V such that S is a subset of S, then
More informationMathematics 205 HWK 6 Solutions Section 13.3 p627. Note: Remember that boldface is being used here, rather than overhead arrows, to indicate vectors.
Mathematics 205 HWK 6 Solutions Section 13.3 p627 Note: Remember that boldface is being used here, rather than overhead arrows, to indicate vectors. Problem 5, 13.3, p627. Given a = 2j + k or a = (0,2,
More informationDefinition: A vector is a directed line segment that has and. Each vector has an initial point and a terminal point.
6.1 Vectors in the Plane PreCalculus 6.1 VECTORS IN THE PLANE Learning Targets: 1. Find the component form and the magnitude of a vector.. Perform addition and scalar multiplication of two vectors. 3.
More informationMath 333  Practice Exam 2 with Some Solutions
Math 333  Practice Exam 2 with Some Solutions (Note that the exam will NOT be this long) Definitions (0 points) Let T : V W be a transformation Let A be a square matrix (a) Define T is linear (b) Define
More informationSection 11.4: Equations of Lines and Planes
Section 11.4: Equations of Lines and Planes Definition: The line containing the point ( 0, 0, 0 ) and parallel to the vector v = A, B, C has parametric equations = 0 + At, = 0 + Bt, = 0 + Ct, where t R
More informationMATH 551  APPLIED MATRIX THEORY
MATH 55  APPLIED MATRIX THEORY FINAL TEST: SAMPLE with SOLUTIONS (25 points NAME: PROBLEM (3 points A web of 5 pages is described by a directed graph whose matrix is given by A Do the following ( points
More information