Section 6.2, 6.3 Orthogonal Sets, Orthogonal Projections

Size: px
Start display at page:

Download "Section 6.2, 6.3 Orthogonal Sets, Orthogonal Projections"

Transcription

1 Section 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 p }; geometry computation interpretation. Expressing y in terms of an OG basis. Projection of y onto subspaces W and W. Orthonormal sets Orthogonal matrices A set of vectors {u... u p } in IR n is an orthogonal set if u i u j = whenever i j. Examples - The set {u u u } = is an OG set because u u = u u = u u =. - The set {v v v } = is not an OG set because v is not OG to v or v.

2 OG LI. Theorem 4 Suppose S = {u... u p } is an OG set of nonzero vectors in IR n. Then S is a linearly independent set. Thus if W = span{u... u p } S is a basis for W. Proof that OG LI Examine the condition for linear independence. That is find all combinations of c i s that make a true statement out of: c u + c u + + c p u p =. Question: OG vectors? What s the difference between LI vectors and An orthogonal basis for a subspace W of IR n is a basis for W that is also an orthogonal set.

3 OG Projection of y onto u; onto {u... u p } Background: Let y u be nonzero unequal vectors in IR n. Express y as y = ŷ + z where ŷ is a vector in the same direction as u and z is perpendicular to u. The vector ŷ in the example above is called the orthogonal projection of y onto u and is denoted proj u y. The vector z in the example above is called the component of y orthogonal to u.

4 Exercises 8. For y = and u = find the orthogonal 4 projection of y onto u. Then write y as the sum of a vector in span{u} and a vector orthogonal to u. 5. For y = 5 that s closest to y. and u = find the point in span{u}. For the problem above find the distance from y to span{u} 4

5 4. Let y = 4 u = 6 6 and u = 4 4. (a) Prove that {u u } is an OG basis for IR. (b) Sketch y u u proj u y and proj u y. (c) What is y (proj u y + proj u y)? Explain. Solution: (a) u u = so u u are OG hence LI. Two LI vectors in IR form a basis of IR. (Could also use Thm. 4) (b) (c) y (proj u y + proj u y) = because projection onto all basis vectors captures every part of y. 5

6 Expressing y in terms of an OG basis Theorem 5 Let {u... u p } be an orthogonal basis for a subspace W of IR n. For each y W the unique weights (c c... c p ) in the linear combination are given by y = c u + + c p u p c j = y u j u j u j for j = p. 5. Express y = 4 in the orthogonal basis as a linear combination of the vectors = {v v v } of IR. Solution: y = proj v y + proj v y + proj v y y v y v = v + v + v v v v = ( /)v + (/)v + ()v y v v v v 6

7 Projection of y onto subspaces W and W 6. Given y = and the OG basis {v v v } = 6 of IR. Define the subspace W by W = span{v v }. Find the component of y that lies within W. [Hint: use theorem 5] 7. For the problem above find the distance from y to W. 7

8 Thm 8: The Orthogonal Decomposition Theorem Let W be a subspace of IR n. Then each y IR n can be written uniquely in the form y = ŷ + z where ŷ W and z W. In fact if {u... u p } is any orthogonal basis of W then ŷ = and z = y ŷ. y u u + u u y u u + + u u y u p u p u p u p Remarks: - The vector ŷ is called the orthogonal projection of y onto W and is denoted by proj W y. - ŷ is the closest vector in W to y. (This is actually Theorem 9 p. 98 in simpler language.) - The distance from y to the subspace W is y ŷ 8

9 Exercise 8. Find the closest point to y in span{u u } where y = 4 u = u =. Solution: The closest point to y in span{u u } is the sum of the components of y along u and u. Thus: closest point = ŷ = y u u u u + y u u u u = 9

10 Orthonormal sets Orthogonal matrices Definitions A set {u... u p } is an orthonormal set if it is an orthogonal set and u i = for i =... p. If W is the subspace spanned by such a set then {u... u p } is an orthonormal basis for W. Example Show that / / / is an orthonormal set. / / = {u u } Solution: - u u = so the vectors are OG. Moreover - u = u u = (/) + ( /) + (/) = 4/9 + 4/9 + /9 = and - u = (/ ) + (/ ) + = / + / =. So {u u } is an orthonormal basis for the subspace (a plane) of IR spanned by this set.

11 Observe: Suppose {u u } is an orthonormal set in IR. Let U be the matrix [ u u ]. Then U T U = u T u T = u u. = u T u u T u u T u u T u Theorem 6 An m n matrix U has orthonormal columns if and only if U T U = I n. Example Recall that orthonormal set. Note that / / / / / / / / / / / / / / / = is an.

12 Other theorems about orthogonal matrices: Theorem 7 Let U be an m n matrix with orthonormal columns and let x and y be in IR n. Then a. Ux = x b. (Ux) (Uy) = x y c. (Ux) (Uy) = if and only if x y = Theorem If {u... u p } is an orthonormal basis for a subspace W of IR n then proj W y = (y u )u + + (y u p )u p. Moreover if U = [ u u u p ] then proj W y = UU T y y IR n. Exercises: see homework

13 Summary Orthogonality LI but LI does not imply orthogonality proj u y = ( ) y u u u u = component of y along u. proj W y = ( ) y u u u u + + y u p u up p u p where {u... u p } is an orthogonal basis of W. [warning this isn t true if the u i are merely LI. They must be OG.] In the above expression u i u i = for i =... p if the basis is orthonormal. Nearest point to y in W distance from y to W Orthogonal matrices U T U = I

MTH 2310, FALL Introduction

MTH 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 information

Orthogonal Complements

Orthogonal 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 information

Math 3191 Applied Linear Algebra

Math 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 information

Math 2331 Linear Algebra

Math 2331 Linear Algebra 6.2 Orthogonal Sets Math 233 Linear Algebra 6.2 Orthogonal Sets Jiwen He Department of Mathematics, University of Houston jiwenhe@math.uh.edu math.uh.edu/ jiwenhe/math233 Jiwen He, University of Houston

More information

Orthonormal Bases; Gram-Schmidt Process; QR-Decomposition

Orthonormal Bases; Gram-Schmidt Process; QR-Decomposition Orthonormal Bases; Gram-Schmidt Process; QR-Decomposition MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 205 Motivation When working with an inner product space, the most

More information

Chapter 6: Orthogonality

Chapter 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 information

Math 3191 Applied Linear Algebra

Math 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 information

Lecture 10: Vector Algebra: Orthogonal Basis

Lecture 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 information

Math 2331 Linear Algebra

Math 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 information

Section 6.4. The Gram Schmidt Process

Section 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 information

Math 261 Lecture Notes: Sections 6.1, 6.2, 6.3 and 6.4 Orthogonal Sets and Projections

Math 261 Lecture Notes: Sections 6.1, 6.2, 6.3 and 6.4 Orthogonal Sets and Projections Math 6 Lecture Notes: Sections 6., 6., 6. and 6. Orthogonal Sets and Projections We will not cover general inner product spaces. We will, however, focus on a particular inner product space the inner product

More information

6.1. Inner Product, Length and Orthogonality

6.1. Inner Product, Length and Orthogonality These are brief notes for the lecture on Friday November 13, and Monday November 1, 2009: they are not complete, but they are a guide to what I want to say on those days. They are guaranteed to be incorrect..1.

More information

Homework 5. (due Wednesday 8 th Nov midnight)

Homework 5. (due Wednesday 8 th Nov midnight) Homework (due Wednesday 8 th Nov midnight) Use this definition for Column Space of a Matrix Column Space of a matrix A is the set ColA of all linear combinations of the columns of A. In other words, if

More information

March 27 Math 3260 sec. 56 Spring 2018

March 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 information

Solutions to Review Problems for Chapter 6 ( ), 7.1

Solutions 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 information

Worksheet for Lecture 25 Section 6.4 Gram-Schmidt Process

Worksheet for Lecture 25 Section 6.4 Gram-Schmidt Process Worksheet for Lecture Name: Section.4 Gram-Schmidt Process Goal For a subspace W = Span{v,..., v n }, we want to find an orthonormal basis of W. Example Let W = Span{x, x } with x = and x =. Give an orthogonal

More information

6. Orthogonality and Least-Squares

6. 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 information

Math Linear Algebra

Math Linear Algebra Math 220 - Linear Algebra (Summer 208) Solutions to Homework #7 Exercise 6..20 (a) TRUE. u v v u = 0 is equivalent to u v = v u. The latter identity is true due to the commutative property of the inner

More information

Announcements Monday, November 20

Announcements 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 information

Linear Models Review

Linear Models Review Linear Models Review Vectors in IR n will be written as ordered n-tuples which are understood to be column vectors, or n 1 matrices. A vector variable will be indicted with bold face, and the prime sign

More information

The Gram Schmidt Process

The Gram Schmidt Process u 2 u The Gram Schmidt Process Now we will present a procedure, based on orthogonal projection, that converts any linearly independent set of vectors into an orthogonal set. Let us begin with the simple

More information

The Gram Schmidt Process

The Gram Schmidt Process The Gram Schmidt Process Now we will present a procedure, based on orthogonal projection, that converts any linearly independent set of vectors into an orthogonal set. Let us begin with the simple case

More information

2.4 Hilbert Spaces. Outline

2.4 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 information

Math 1180, Notes, 14 1 C. v 1 v n v 2. C A ; w n. A and w = v i w i : v w = i=1

Math 1180, Notes, 14 1 C. v 1 v n v 2. C A ; w n. A and w = v i w i : v w = i=1 Math 8, 9 Notes, 4 Orthogonality We now start using the dot product a lot. v v = v v n then by Recall that if w w ; w n and w = v w = nx v i w i : Using this denition, we dene the \norm", or length, of

More information

MATH 304 Linear Algebra Lecture 20: The Gram-Schmidt process (continued). Eigenvalues and eigenvectors.

MATH 304 Linear Algebra Lecture 20: The Gram-Schmidt process (continued). Eigenvalues and eigenvectors. MATH 304 Linear Algebra Lecture 20: The Gram-Schmidt process (continued). Eigenvalues and eigenvectors. Orthogonal sets Let V be a vector space with an inner product. Definition. Nonzero vectors v 1,v

More information

LINEAR ALGEBRA SUMMARY SHEET.

LINEAR 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 information

MTH 2032 SemesterII

MTH 2032 SemesterII MTH 202 SemesterII 2010-11 Linear Algebra Worked Examples Dr. Tony Yee Department of Mathematics and Information Technology The Hong Kong Institute of Education December 28, 2011 ii Contents Table of Contents

More information

Recall: Dot product on R 2 : u v = (u 1, u 2 ) (v 1, v 2 ) = u 1 v 1 + u 2 v 2, u u = u u 2 2 = u 2. Geometric Meaning:

Recall: Dot product on R 2 : u v = (u 1, u 2 ) (v 1, v 2 ) = u 1 v 1 + u 2 v 2, u u = u u 2 2 = u 2. Geometric Meaning: Recall: Dot product on R 2 : u v = (u 1, u 2 ) (v 1, v 2 ) = u 1 v 1 + u 2 v 2, u u = u 2 1 + u 2 2 = u 2. Geometric Meaning: u v = u v cos θ. u θ v 1 Reason: The opposite side is given by u v. u v 2 =

More information

MATH 20F: LINEAR ALGEBRA LECTURE B00 (T. KEMP)

MATH 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 information

Problem # Max points possible Actual score Total 120

Problem # 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 information

Linear Systems. Class 27. c 2008 Ron Buckmire. TITLE Projection Matrices and Orthogonal Diagonalization CURRENT READING Poole 5.4

Linear Systems. Class 27. c 2008 Ron Buckmire. TITLE Projection Matrices and Orthogonal Diagonalization CURRENT READING Poole 5.4 Linear Systems Math Spring 8 c 8 Ron Buckmire Fowler 9 MWF 9: am - :5 am http://faculty.oxy.edu/ron/math//8/ Class 7 TITLE Projection Matrices and Orthogonal Diagonalization CURRENT READING Poole 5. Summary

More information

Orthogonality. Orthonormal Bases, Orthogonal Matrices. Orthogonality

Orthogonality. Orthonormal Bases, Orthogonal Matrices. Orthogonality Orthonormal Bases, Orthogonal Matrices The Major Ideas from Last Lecture Vector Span Subspace Basis Vectors Coordinates in different bases Matrix Factorization (Basics) The Major Ideas from Last Lecture

More information

Orthogonality and Least Squares

Orthogonality 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 information

The Gram-Schmidt Process 1

The Gram-Schmidt Process 1 The Gram-Schmidt Process In this section all vector spaces will be subspaces of some R m. Definition.. Let S = {v...v n } R m. The set S is said to be orthogonal if v v j = whenever i j. If in addition

More information

Dot product and linear least squares problems

Dot product and linear least squares problems Dot product and linear least squares problems Dot Product For vectors u,v R n we define the dot product Note that we can also write this as u v = u,,u n u v = u v + + u n v n v v n = u v + + u n v n The

More information

Math 416, Spring 2010 Gram-Schmidt, the QR-factorization, Orthogonal Matrices March 4, 2010 GRAM-SCHMIDT, THE QR-FACTORIZATION, ORTHOGONAL MATRICES

Math 416, Spring 2010 Gram-Schmidt, the QR-factorization, Orthogonal Matrices March 4, 2010 GRAM-SCHMIDT, THE QR-FACTORIZATION, ORTHOGONAL MATRICES Math 46, Spring 00 Gram-Schmidt, the QR-factorization, Orthogonal Matrices March 4, 00 GRAM-SCHMIDT, THE QR-FACTORIZATION, ORTHOGONAL MATRICES Recap Yesterday we talked about several new, important concepts

More information

Announcements Monday, November 26

Announcements 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 information

Vector Spaces, Orthogonality, and Linear Least Squares

Vector Spaces, Orthogonality, and Linear Least Squares Week Vector Spaces, Orthogonality, and Linear Least Squares. Opening Remarks.. Visualizing Planes, Lines, and Solutions Consider the following system of linear equations from the opener for Week 9: χ χ

More information

orthogonal relations between vectors and subspaces Then we study some applications in vector spaces and linear systems, including Orthonormal Basis,

orthogonal 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

Designing Information Devices and Systems II

Designing Information Devices and Systems II EECS 16B Fall 2016 Designing Information Devices and Systems II Linear Algebra Notes Introduction In this set of notes, we will derive the linear least squares equation, study the properties symmetric

More information

A Primer in Econometric Theory

A Primer in Econometric Theory A Primer in Econometric Theory Lecture 1: Vector Spaces John Stachurski Lectures by Akshay Shanker May 5, 2017 1/104 Overview Linear algebra is an important foundation for mathematics and, in particular,

More information

Fall TMA4145 Linear Methods. Exercise set Given the matrix 1 2

Fall TMA4145 Linear Methods. Exercise set Given the matrix 1 2 Norwegian University of Science and Technology Department of Mathematical Sciences TMA445 Linear Methods Fall 07 Exercise set Please justify your answers! The most important part is how you arrive at an

More information

IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET

IMPORTANT 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 information

1 0 1, then use that decomposition to solve the least squares problem. 1 Ax = 2. q 1 = a 1 a 1 = 1. to find the intermediate result:

1 0 1, then use that decomposition to solve the least squares problem. 1 Ax = 2. q 1 = a 1 a 1 = 1. to find the intermediate result: Exercise Find the QR decomposition of A =, then use that decomposition to solve the least squares problem Ax = 2 3 4 Solution Name the columns of A by A = [a a 2 a 3 ] and denote the columns of the results

More information

Math 18, Linear Algebra, Lecture C00, Spring 2017 Review and Practice Problems for Final Exam

Math 18, Linear Algebra, Lecture C00, Spring 2017 Review and Practice Problems for Final Exam Math 8, Linear Algebra, Lecture C, Spring 7 Review and Practice Problems for Final Exam. The augmentedmatrix of a linear system has been transformed by row operations into 5 4 8. Determine if the system

More information

Assignment #9: Orthogonal Projections, Gram-Schmidt, and Least Squares. Name:

Assignment #9: Orthogonal Projections, Gram-Schmidt, and Least Squares. Name: Assignment 9: Orthogonal Projections, Gram-Schmidt, and Least Squares Due date: Friday, April 0, 08 (:pm) Name: Section Number Assignment 9: Orthogonal Projections, Gram-Schmidt, and Least Squares Due

More information

MATH Linear Algebra

MATH Linear Algebra MATH 304 - Linear Algebra In the previous note we learned an important algorithm to produce orthogonal sequences of vectors called the Gramm-Schmidt orthogonalization process. Gramm-Schmidt orthogonalization

More information

3. For a given dataset and linear model, what do you think is true about least squares estimates? Is Ŷ always unique? Yes. Is ˆβ always unique? No.

3. For a given dataset and linear model, what do you think is true about least squares estimates? Is Ŷ always unique? Yes. Is ˆβ always unique? No. 7. LEAST SQUARES ESTIMATION 1 EXERCISE: Least-Squares Estimation and Uniqueness of Estimates 1. For n real numbers a 1,...,a n, what value of a minimizes the sum of squared distances from a to each of

More information

Algorithms to Compute Bases and the Rank of a Matrix

Algorithms to Compute Bases and the Rank of a Matrix Algorithms to Compute Bases and the Rank of a Matrix Subspaces associated to a matrix Suppose that A is an m n matrix The row space of A is the subspace of R n spanned by the rows of A The column space

More information

Lecture 13: Orthogonal projections and least squares (Section ) Thang Huynh, UC San Diego 2/9/2018

Lecture 13: Orthogonal projections and least squares (Section ) Thang Huynh, UC San Diego 2/9/2018 Lecture 13: Orthogonal projections and least squares (Section 3.2-3.3) Thang Huynh, UC San Diego 2/9/2018 Orthogonal projection onto subspaces Theorem. Let W be a subspace of R n. Then, each x in R n can

More information

Chapter 6. Orthogonality and Least Squares

Chapter 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 information

MATH 221: SOLUTIONS TO SELECTED HOMEWORK PROBLEMS

MATH 221: SOLUTIONS TO SELECTED HOMEWORK PROBLEMS MATH 221: SOLUTIONS TO SELECTED HOMEWORK PROBLEMS 1. HW 1: Due September 4 1.1.21. Suppose v, w R n and c is a scalar. Prove that Span(v + cw, w) = Span(v, w). We must prove two things: that every element

More information

IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET

IMPORTANT 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 information

Chapter 4 Euclid Space

Chapter 4 Euclid Space Chapter 4 Euclid Space Inner Product Spaces Definition.. Let V be a real vector space over IR. A real inner product on V is a real valued function on V V, denoted by (, ), which satisfies () (x, y) = (y,

More information

and u and v are orthogonal if and only if u v = 0. u v = x1x2 + y1y2 + z1z2. 1. In R 3 the dot product is defined by

and u and v are orthogonal if and only if u v = 0. u v = x1x2 + y1y2 + z1z2. 1. In R 3 the dot product is defined by Linear Algebra [] 4.2 The Dot Product and Projections. In R 3 the dot product is defined by u v = u v cos θ. 2. For u = (x, y, z) and v = (x2, y2, z2), we have u v = xx2 + yy2 + zz2. 3. cos θ = u v u v,

More information

Homework 11 Solutions. Math 110, Fall 2013.

Homework 11 Solutions. Math 110, Fall 2013. Homework 11 Solutions Math 110, Fall 2013 1 a) Suppose that T were self-adjoint Then, the Spectral Theorem tells us that there would exist an orthonormal basis of P 2 (R), (p 1, p 2, p 3 ), consisting

More information

Inner products. Theorem (basic properties): Given vectors u, v, w in an inner product space V, and a scalar k, the following properties hold:

Inner products. Theorem (basic properties): Given vectors u, v, w in an inner product space V, and a scalar k, the following properties hold: Inner products Definition: An inner product on a real vector space V is an operation (function) that assigns to each pair of vectors ( u, v) in V a scalar u, v satisfying the following axioms: 1. u, v

More information

Announcements Monday, November 19

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 information

Row Space, Column Space, and Nullspace

Row Space, Column Space, and Nullspace Row Space, Column Space, and Nullspace MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Introduction Every matrix has associated with it three vector spaces: row space

More information

STAT 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. 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 information

Miderm II Solutions To find the inverse we row-reduce the augumented matrix [I A]. In our case, we row reduce

Miderm II Solutions To find the inverse we row-reduce the augumented matrix [I A]. In our case, we row reduce Miderm II Solutions Problem. [8 points] (i) [4] Find the inverse of the matrix A = To find the inverse we row-reduce the augumented matrix [I A]. In our case, we row reduce We have A = 2 2 (ii) [2] Possibly

More information

Section 6.1. Inner Product, Length, and Orthogonality

Section 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 information

Lecture 03. Math 22 Summer 2017 Section 2 June 26, 2017

Lecture 03. Math 22 Summer 2017 Section 2 June 26, 2017 Lecture 03 Math 22 Summer 2017 Section 2 June 26, 2017 Just for today (10 minutes) Review row reduction algorithm (40 minutes) 1.3 (15 minutes) Classwork Review row reduction algorithm Review row reduction

More information

Lecture 4 Orthonormal vectors and QR factorization

Lecture 4 Orthonormal vectors and QR factorization Orthonormal vectors and QR factorization 4 1 Lecture 4 Orthonormal vectors and QR factorization EE263 Autumn 2004 orthonormal vectors Gram-Schmidt procedure, QR factorization orthogonal decomposition induced

More information

Chapter 7: Symmetric Matrices and Quadratic Forms

Chapter 7: Symmetric Matrices and Quadratic Forms Chapter 7: Symmetric Matrices and Quadratic Forms (Last Updated: December, 06) These notes are derived primarily from Linear Algebra and its applications by David Lay (4ed). A few theorems have been moved

More information

Announcements Monday, November 26

Announcements 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 information

P = A(A T A) 1 A T. A Om (m n)

P = A(A T A) 1 A T. A Om (m n) Chapter 4: Orthogonality 4.. Projections Proposition. Let A be a matrix. Then N(A T A) N(A). Proof. If Ax, then of course A T Ax. Conversely, if A T Ax, then so Ax also. x (A T Ax) x T A T Ax (Ax) T Ax

More information

18.06 Professor Johnson Quiz 1 October 3, 2007

18.06 Professor Johnson Quiz 1 October 3, 2007 18.6 Professor Johnson Quiz 1 October 3, 7 SOLUTIONS 1 3 pts.) A given circuit network directed graph) which has an m n incidence matrix A rows = edges, columns = nodes) and a conductance matrix C [diagonal

More information

Lecture 3: Linear Algebra Review, Part II

Lecture 3: Linear Algebra Review, Part II Lecture 3: Linear Algebra Review, Part II Brian Borchers January 4, Linear Independence Definition The vectors v, v,..., v n are linearly independent if the system of equations c v + c v +...+ c n v n

More information

There are two things that are particularly nice about the first basis

There are two things that are particularly nice about the first basis Orthogonality and the Gram-Schmidt Process In Chapter 4, we spent a great deal of time studying the problem of finding a basis for a vector space We know that a basis for a vector space can potentially

More information

Orthogonality. 6.1 Orthogonal Vectors and Subspaces. Chapter 6

Orthogonality. 6.1 Orthogonal Vectors and Subspaces. Chapter 6 Chapter 6 Orthogonality 6.1 Orthogonal Vectors and Subspaces Recall that if nonzero vectors x, y R n are linearly independent then the subspace of all vectors αx + βy, α, β R (the space spanned by x and

More information

Chapter 6 - Orthogonality

Chapter 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 information

Chapter 6. Orthogonality

Chapter 6. Orthogonality 6.4 The Projection Matrix 1 Chapter 6. Orthogonality 6.4 The Projection Matrix Note. In Section 6.1 (Projections), we projected a vector b R n onto a subspace W of R n. We did so by finding a basis for

More information

MAT2342 : Introduction to Applied Linear Algebra Mike Newman, fall Projections. introduction

MAT2342 : Introduction to Applied Linear Algebra Mike Newman, fall Projections. introduction MAT4 : Introduction to Applied Linear Algebra Mike Newman fall 7 9. Projections introduction One reason to consider projections is to understand approximate solutions to linear systems. A common example

More information

Math Linear Algebra II. 1. Inner Products and Norms

Math 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 information

Math 550 Notes. Chapter 2. Jesse Crawford. Department of Mathematics Tarleton State University. Fall 2010

Math 550 Notes. Chapter 2. Jesse Crawford. Department of Mathematics Tarleton State University. Fall 2010 Math 550 Notes Chapter 2 Jesse Crawford Department of Mathematics Tarleton State University Fall 2010 (Tarleton State University) Math 550 Chapter 2 Fall 2010 1 / 20 Linear algebra deals with finite dimensional

More information

WI1403-LR Linear Algebra. Delft University of Technology

WI1403-LR Linear Algebra. Delft University of Technology WI1403-LR Linear Algebra Delft University of Technology Year 2013 2014 Michele Facchinelli Version 10 Last modified on February 1, 2017 Preface This summary was written for the course WI1403-LR Linear

More information

which are not all zero. The proof in the case where some vector other than combination of the other vectors in S is similar.

which are not all zero. The proof in the case where some vector other than combination of the other vectors in S is similar. It follows that S is linearly dependent since the equation is satisfied by which are not all zero. The proof in the case where some vector other than combination of the other vectors in S is similar. is

More information

2. Every linear system with the same number of equations as unknowns has a unique solution.

2. Every linear system with the same number of equations as unknowns has a unique solution. 1. For matrices A, B, C, A + B = A + C if and only if A = B. 2. Every linear system with the same number of equations as unknowns has a unique solution. 3. Every linear system with the same number of equations

More information

Solutions to Math 51 Midterm 1 July 6, 2016

Solutions to Math 51 Midterm 1 July 6, 2016 Solutions to Math 5 Midterm July 6, 26. (a) (6 points) Find an equation (of the form ax + by + cz = d) for the plane P in R 3 passing through the points (, 2, ), (2,, ), and (,, ). We first compute two

More information

Lecture 1: Basic Concepts

Lecture 1: Basic Concepts ENGG 5781: Matrix Analysis and Computations Lecture 1: Basic Concepts 2018-19 First Term Instructor: Wing-Kin Ma This note is not a supplementary material for the main slides. I will write notes such as

More information

For each problem, place the letter choice of your answer in the spaces provided on this page.

For each problem, place the letter choice of your answer in the spaces provided on this page. Math 6 Final Exam Spring 6 Your name Directions: For each problem, place the letter choice of our answer in the spaces provided on this page...... 6. 7. 8. 9....... 6. 7. 8. 9....... B signing here, I

More information

No books, no notes, no calculators. You must show work, unless the question is a true/false, yes/no, or fill-in-the-blank question.

No books, no notes, no calculators. You must show work, unless the question is a true/false, yes/no, or fill-in-the-blank question. Math 304 Final Exam (May 8) Spring 206 No books, no notes, no calculators. You must show work, unless the question is a true/false, yes/no, or fill-in-the-blank question. Name: Section: Question Points

More information

GENERAL VECTOR SPACES AND SUBSPACES [4.1]

GENERAL VECTOR SPACES AND SUBSPACES [4.1] GENERAL VECTOR SPACES AND SUBSPACES [4.1] General vector spaces So far we have seen special spaces of vectors of n dimensions denoted by R n. It is possible to define more general vector spaces A vector

More information

SOLUTIONS TO EXERCISES FOR MATHEMATICS 133 Part 1. I. Topics from linear algebra

SOLUTIONS TO EXERCISES FOR MATHEMATICS 133 Part 1. I. Topics from linear algebra SOLUTIONS TO EXERCISES FOR MATHEMATICS 133 Part 1 Winter 2009 I. Topics from linear algebra I.0 : Background 1. Suppose that {x, y} is linearly dependent. Then there are scalars a, b which are not both

More information

Assignment 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. 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 information

Dot Products. K. Behrend. April 3, Abstract A short review of some basic facts on the dot product. Projections. The spectral theorem.

Dot Products. K. Behrend. April 3, Abstract A short review of some basic facts on the dot product. Projections. The spectral theorem. Dot Products K. Behrend April 3, 008 Abstract A short review of some basic facts on the dot product. Projections. The spectral theorem. Contents The dot product 3. Length of a vector........................

More information

Math 102, Winter 2009, Homework 7

Math 102, Winter 2009, Homework 7 Math 2, Winter 29, Homework 7 () Find the standard matrix of the linear transformation T : R 3 R 3 obtained by reflection through the plane x + z = followed by a rotation about the positive x-axes by 6

More information

Warm-up. True or false? Baby proof. 2. The system of normal equations for A x = y has solutions iff A x = y has solutions

Warm-up. True or false? Baby proof. 2. The system of normal equations for A x = y has solutions iff A x = y has solutions Warm-up True or false? 1. proj u proj v u = u 2. The system of normal equations for A x = y has solutions iff A x = y has solutions 3. The normal equations are always consistent Baby proof 1. Let A be

More information

MATH 22A: LINEAR ALGEBRA Chapter 4

MATH 22A: LINEAR ALGEBRA Chapter 4 MATH 22A: LINEAR ALGEBRA Chapter 4 Jesús De Loera, UC Davis November 30, 2012 Orthogonality and Least Squares Approximation QUESTION: Suppose Ax = b has no solution!! Then what to do? Can we find an Approximate

More information

DS-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 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 information

MATH 1120 (LINEAR ALGEBRA 1), FINAL EXAM FALL 2011 SOLUTIONS TO PRACTICE VERSION

MATH 1120 (LINEAR ALGEBRA 1), FINAL EXAM FALL 2011 SOLUTIONS TO PRACTICE VERSION MATH (LINEAR ALGEBRA ) FINAL EXAM FALL SOLUTIONS TO PRACTICE VERSION Problem (a) For each matrix below (i) find a basis for its column space (ii) find a basis for its row space (iii) determine whether

More information

MTH 309Y 37. Inner product spaces. = a 1 b 1 + a 2 b a n b n

MTH 309Y 37. Inner product spaces. = a 1 b 1 + a 2 b a n b n MTH 39Y 37. Inner product spaces Recall: ) The dot product in R n : a. a n b. b n = a b + a 2 b 2 +...a n b n 2) Properties of the dot product: a) u v = v u b) (u + v) w = u w + v w c) (cu) v = c(u v)

More information

University of Colorado Denver Department of Mathematical and Statistical Sciences Applied Linear Algebra Ph.D. Preliminary Exam June 8, 2012

University of Colorado Denver Department of Mathematical and Statistical Sciences Applied Linear Algebra Ph.D. Preliminary Exam June 8, 2012 University of Colorado Denver Department of Mathematical and Statistical Sciences Applied Linear Algebra Ph.D. Preliminary Exam June 8, 2012 Name: Exam Rules: This is a closed book exam. Once the exam

More information

(a) If A is a 3 by 4 matrix, what does this tell us about its nullspace? Solution: dim N(A) 1, since rank(a) 3. Ax =

(a) If A is a 3 by 4 matrix, what does this tell us about its nullspace? Solution: dim N(A) 1, since rank(a) 3. Ax = . (5 points) (a) If A is a 3 by 4 matrix, what does this tell us about its nullspace? dim N(A), since rank(a) 3. (b) If we also know that Ax = has no solution, what do we know about the rank of A? C(A)

More information

This property turns out to be a general property of eigenvectors of a symmetric A that correspond to distinct eigenvalues as we shall see later.

This property turns out to be a general property of eigenvectors of a symmetric A that correspond to distinct eigenvalues as we shall see later. 34 To obtain an eigenvector x 2 0 2 for l 2 = 0, define: B 2 A - l 2 I 2 = È 1, 1, 1 Î 1-0 È 1, 0, 0 Î 1 = È 1, 1, 1 Î 1. To transform B 2 into an upper triangular matrix, subtract the first row of B 2

More information

Typical Problem: Compute.

Typical Problem: Compute. Math 2040 Chapter 6 Orhtogonality and Least Squares 6.1 and some of 6.7: Inner Product, Length and Orthogonality. Definition: If x, y R n, then x y = x 1 y 1 +... + x n y n is the dot product of x and

More information

1. Diagonalize the matrix A if possible, that is, find an invertible matrix P and a diagonal

1. Diagonalize the matrix A if possible, that is, find an invertible matrix P and a diagonal . Diagonalize the matrix A if possible, that is, find an invertible matrix P and a diagonal 3 9 matrix D such that A = P DP, for A =. 3 4 3 (a) P = 4, D =. 3 (b) P = 4, D =. (c) P = 4 8 4, D =. 3 (d) P

More information

The set of all solutions to the homogeneous equation Ax = 0 is a subspace of R n if A is m n.

The set of all solutions to the homogeneous equation Ax = 0 is a subspace of R n if A is m n. 0 Subspaces (Now, we are ready to start the course....) Definitions: A linear combination of the vectors v, v,..., v m is any vector of the form c v + c v +... + c m v m, where c,..., c m R. A subset V

More information

Orthogonal complement

Orthogonal complement Orthogonal complement Aim lecture: Inner products give a special way of constructing vector space complements. As usual, in this lecture F = R or C. We also let V be an F-space equipped with an inner product

More information