# Lesson 14 The Gram-Schmidt Process and QR Factorizations

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Lesson 14 The Gram-Schmidt Process and QR Factorizations Math 21b March 12, 2007 Announcements Get your midterm from me if you haven t yet. Homework for March 14: 5.2: 6,14,22,34,40,44* Problem Session Wednesdays, 7-8 PM in SC 101b Office hours: Monday 2-4, Tuesday 3-5 in SC 323 (resuming today)

2 The problem Given a basis B for a subspace V of R n, replace it with an orthonormal basis A having the same span.

3 The problem Given a basis B for a subspace V of R n, replace it with an orthonormal basis A having the same span. Why? Because orthonormal bases are just plain better.

4 Why orthonormal bases are just plain better No need to check for linear independence (automatic from orthogonality) Coordinates are easy just dot: v u 1 v u 2 [ v] A =. v u m Orthogonal projections are easy just dot: proj V ( x) = x = ( x u 1 ) u 1 + ( x u 2 ) u ( x u m ) u m

5 The geometric idea In fact, we will do a little better: Given a basis v 1, v 2,..., v m for a subspace V of R n, replace it with an orthonormal basis u 1, u 2,..., u m such that span( v 1, v 2,..., v k ) = span( u 1, u 2,..., u k ) for each k between 1 and m.

6 The geometric idea In fact, we will do a little better: Given a basis v 1, v 2,..., v m for a subspace V of R n, replace it with an orthonormal basis u 1, u 2,..., u m such that span( v 1, v 2,..., v k ) = span( u 1, u 2,..., u k ) for each k between 1 and m. The process works like this: Scale v 1 to make a unit vector u 1. Choose the rest of the u k to be the (normalized) perpendicular part of the orthogonal decomposition of v k relative to span( v 1, v 2,..., v k 1 ).

7 The geometric idea, in pictures v 2 v 1

8 The geometric idea, in pictures v 2 u 1 v 1

9 The geometric idea, in pictures v 2 u 1 v 1

10 The geometric idea, in pictures v 2 v 2 v 1 v 2 u 1

11 The geometric idea, in pictures v 2 v 2 u 2 v 1 v 2 u 1

12 The geometric idea, in pictures v 2 u 2 v 1 u 1

13 The geometric idea, in 3D v 3 v 2 v 1

14 The geometric idea, in 3D v 3 v 1 u 1 v 2

15 The geometric idea, in 3D v 3 v 1 u 1 v 2

16 The geometric idea, in 3D v 2 v 3 v 1 u 1 v 2 v 2

17 The geometric idea, in 3D v 2 v 3 u 2 v 1 u 1 v 2 v 2

18 The geometric idea, in 3D v 3 u 2 v 2 v 1 u 1

19 The geometric idea, in 3D v 3 v 3 u 2 v 2 v 1 u 1 v 3

20 The geometric idea, in 3D v 3 v 3 u 3 u 2 v 2 v 1 u 1 v 3

21 The geometric idea, in 3D v 3 u3 u 2 v 2 v 1 u 1

22 The algebraic idea Scale v 1 to make a unit vector u 1.

23 The algebraic idea Scale v 1 to make a unit vector u 1. Take v 2 to be the perpendicular part in the orthogonal decomposition of v 2 relative to the subspace spanned by v 1 : v 2 = v 2 proj v1 v 2 = v 2 ( u 1 v 2 ) u 1

24 The algebraic idea Scale v 1 to make a unit vector u 1. Take v 2 to be the perpendicular part in the orthogonal decomposition of v 2 relative to the subspace spanned by v 1 : v 2 = v 2 proj v1 v 2 = v 2 ( u 1 v 2 ) u 1 Scale v 2 to make a unit vector u 2.

25 The algebraic idea Scale v 1 to make a unit vector u 1. Take v 2 to be the perpendicular part in the orthogonal decomposition of v 2 relative to the subspace spanned by v 1 : v 2 = v 2 proj v1 v 2 = v 2 ( u 1 v 2 ) u 1 Scale v 2 to make a unit vector u 2. Take v 3 to be the perpendicular part in the orthogonal decomposition of v 3 relative to the subspace spanned by v 1 and v 2 : v 3 = v 3 proj v1, v 2 v 3 = v 3 ( u 1 v 3 ) u 1 ( u 2 v 3 ) u 2

26 The algebraic idea Scale v 1 to make a unit vector u 1. Take v 2 to be the perpendicular part in the orthogonal decomposition of v 2 relative to the subspace spanned by v 1 : v 2 = v 2 proj v1 v 2 = v 2 ( u 1 v 2 ) u 1 Scale v 2 to make a unit vector u 2. Take v 3 to be the perpendicular part in the orthogonal decomposition of v 3 relative to the subspace spanned by v 1 and v 2 : v 3 = v 3 proj v1, v 2 v 3 = v 3 ( u 1 v 3 ) u 1 ( u 2 v 3 ) u 2 Scale v 3 to make a unit vector u

27 The algebraic idea Scale v 1 to make a unit vector u 1. Take v 2 to be the perpendicular part in the orthogonal decomposition of v 2 relative to the subspace spanned by v 1 : v 2 = v 2 proj v1 v 2 = v 2 ( u 1 v 2 ) u 1 Scale v 2 to make a unit vector u 2. Take v 3 to be the perpendicular part in the orthogonal decomposition of v 3 relative to the subspace spanned by v 1 and v 2 : v 3 = v 3 proj v1, v 2 v 3 = v 3 ( u 1 v 3 ) u 1 ( u 2 v 3 ) u 2 Scale v 3 to make a unit vector u

28 The algebraic idea Scale v 1 to make a unit vector u 1. Take v 2 to be the perpendicular part in the orthogonal decomposition of v 2 relative to the subspace spanned by v 1 : v 2 = v 2 proj v1 v 2 = v 2 ( u 1 v 2 ) u 1 Scale v 2 to make a unit vector u 2. Take v 3 to be the perpendicular part in the orthogonal decomposition of v 3 relative to the subspace spanned by v 1 and v 2 : v 3 = v 3 proj v1, v 2 v 3 = v 3 ( u 1 v 3 ) u 1 ( u 2 v 3 ) u 2 Scale v 3 to make a unit vector u This is known as the Gram-Schmidt process for orthonormalization.

29 Worksheet Do Problems 1 3

30 QR Factorizations

31 When are the spans of two sets the same? When is span( v 1, v 2,..., v m ) = span( u 1, u 2,..., u m )?

32 When are the spans of two sets the same? When is span( v 1, v 2,..., v m ) = span( u 1, u 2,..., u m )? If each of the v i s can be written as a linear combination of the u i s and vice versa. v 1 = c 11 u 1 + c 21 u c m1 u m v 2 = c 12 u 1 + c 22 u c m2 u m. v m = c 1m u 1 + c 2m u c mm u m

33 Writing it all in terms of matrices, we have [ v1 c 11 c 12 c 1m ] [ ] c 21 c 22 c 2m v 2... v m = u1 u 2... u m c m1 } c m2 {{ c mm } S

34 Writing it all in terms of matrices, we have [ v1 c 11 c 12 c 1m ] [ ] c 21 c 22 c 2m v 2... v m = u1 u 2... u m c m1 } c m2 {{ c mm } S The matrix S is the change of basis matrix from ( v 1,..., v m ) to ( u 1,..., u m )

35 From Gram-Schmidt to QR Suppose an n m matrix M has rank m, so its columns v 1,..., v m are linearly independent. Let u 1,..., u m be the orthonormal basis of span( v 1,..., v m ) = image M gotten by the Gram-Schmidt process, and let Q = [ u 1... u m ]. Then M = QR for some invertible matrix R.

36 What is R? u 1 = v 1 v 1 = v 1 = v 1 u 1

37 What is R? u 1 = v 1 v 1 = v 1 = v 1 u 1 v 2 = v 2 ( v 2 u 1 ) u 1 and u 2 = v 2 v 2, so v 2 = ( u 1 v 2 ) u 1 + v 2 u 2

38 What is R? u 1 = v 1 v 1 = v 1 = v 1 u 1 v 2 = v 2 ( v 2 u 1 ) u 1 and u 2 = v 2 v 2, so v 2 = ( u 1 v 2 ) u 1 + v 2 u 2 And so on: v k = ( u 1 v k ) u 1 + ( u 2 v k ) u ( u k 1 v k ) u k 1 + v k u k

39 So v 1 u 1 v 2 u 1 v 3 u 1 v m 0 v 2 u 2 v 3 u 2 v m R = 0 0 v 3... u 3 v m v m That is, R is upper triangular. u i v j if i < j r ij = v i = ui v i if i = j 0 if i > j

40 Example Find the QR factorization of M = [ ]

41 Example Find the QR factorization of M = Solution We have v 1 = [ ] [ ] 1 and r 1 11 = v 1 = 2, so u 2 = [ 1/ ] 2. 1/ 2

42 Example Find the QR factorization of M = Solution We have v 1 = [ ] [ ] 1 and r 1 11 = v 1 = 2, so u 2 = v 2 = v 2 ( u 1 v 2 }{{} r 12 ) u 1 = [ ] [ 1/ ] 2 = 2 1/ 2 [ 1/ ] 2. Also, 1/ 2 [ 1/2 1 /2 ]

43 Example Find the QR factorization of M = Solution We have v 1 = Therefore [ ] [ ] 1 and r 1 11 = v 1 = 2, so u 2 = v 2 = v 2 ( u 1 v 2 }{{} r 12 ) u 1 = r 22 = v 2 [ ] [ 1/ ] 2 = 2 1/ 2 [ 1/ ] 2. Also, 1/ 2 [ 1/2 1 /2 = 1 u 2 = 1 [ 1/ ] v 2 2 = 2 r 22 1 /. 2 ]

44 Example Find the QR factorization of M = Solution We have v 1 = Therefore [ ] [ ] 1 and r 1 11 = v 1 = 2, so u 2 = v 2 = v 2 ( u 1 v 2 }{{} r 12 ) u 1 = r 22 = v 2 [ ] [ 1/ ] 2 = 2 1/ 2 [ 1/ ] 2. Also, 1/ 2 [ 1/2 1 /2 = 1 u 2 = 1 [ 1/ ] v 2 2 = 2 r 22 1 /. 2 Putting this all together, [ ] [ 1 1 1/ 2 1/ ] [ 2 2 1/ ] M = = 1 0 1/ 2 1 / / = QR 2 ]

45 Example (Worksheet, #4) Find the QR Factorization of

46 Example (Worksheet, #4) Find the QR Factorization of Solution 1/2 1/2 1 /2 1 / /2 1/2 Q = 1 /2 1 /2 1/2 1/2 1/2 1/2 1/2 R = / /2 1/2 1 /2 1/2 1 / /2

47 Who cares? A lot of important things are easier to calculate once you have the QR factorization, especially if M is square. It turns out Q 1 = Q T, so M 1 = R 1 Q T and R 1 is easier to calculate than M 1. It turns out det Q = ±1, so det M = ± det R, and det R is very easy to calculate (more later) The eigenvalues of M are easier to calculate when M is factored as QR.

### Similarity 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

### Orthogonal 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

### Chapter 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

### 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

### Applied 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

### 1 2 3 1 1 2 x = + x 2 + x 4 1 0 1

(d) If the vector b is the sum of the four columns of A, write down the complete solution to Ax = b. 1 2 3 1 1 2 x = + x 2 + x 4 1 0 0 1 0 1 2. (11 points) This problem finds the curve y = C + D 2 t which

### Linear 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

### Notes 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,

### LINEAR ALGEBRA. September 23, 2010

LINEAR ALGEBRA September 3, 00 Contents 0. LU-decomposition.................................... 0. Inverses and Transposes................................. 0.3 Column Spaces and NullSpaces.............................

### Inner 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

### MATH 240 Fall, Chapter 1: Linear Equations and Matrices

MATH 240 Fall, 2007 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 9th Ed. written by Prof. J. Beachy Sections 1.1 1.5, 2.1 2.3, 4.2 4.9, 3.1 3.5, 5.3 5.5, 6.1 6.3, 6.5, 7.1 7.3 DEFINITIONS

### CS3220 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

### 1 Eigenvalues and Eigenvectors

Math 20 Chapter 5 Eigenvalues and Eigenvectors Eigenvalues and Eigenvectors. Definition: A scalar λ is called an eigenvalue of the n n matrix A is there is a nontrivial solution x of Ax = λx. Such an x

### 5.3 ORTHOGONAL TRANSFORMATIONS AND ORTHOGONAL MATRICES

5.3 ORTHOGONAL TRANSFORMATIONS AND ORTHOGONAL MATRICES Definition 5.3. Orthogonal transformations and orthogonal matrices A linear transformation T from R n to R n is called orthogonal if it preserves

### Section 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

### [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:

### Lectures 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 real-valued matrix A is said to be an orthogonal

### Recall 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

### is 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 two-dimensional subspace of R. We can describe a vector in this plane by its spatial (D)coordinates; for example, vector x 5

### Inner Product Spaces and Orthogonality

Inner Product Spaces and Orthogonality week 3-4 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,

### Linear Least Squares

Linear Least Squares Suppose we are given a set of data points {(x i,f i )}, i = 1,...,n. These could be measurements from an experiment or obtained simply by evaluating a function at some points. One

### 5. Orthogonal matrices

L Vandenberghe EE133A (Spring 2016) 5 Orthogonal matrices matrices with orthonormal columns orthogonal matrices tall matrices with orthonormal columns complex matrices with orthonormal columns 5-1 Orthonormal

### Review Jeopardy. Blue vs. Orange. Review Jeopardy

Review Jeopardy Blue vs. Orange Review Jeopardy Jeopardy Round Lectures 0-3 Jeopardy Round \$200 How could I measure how far apart (i.e. how different) two observations, y 1 and y 2, are from each other?

### x + 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

### Au = = = 3u. Aw = = = 2w. so the action of A on u and w is very easy to picture: it simply amounts to a stretching by 3 and 2, respectively.

Chapter 7 Eigenvalues and Eigenvectors In this last chapter of our exploration of Linear Algebra we will revisit eigenvalues and eigenvectors of matrices, concepts that were already introduced in Geometry

### 1. True/False: Circle the correct answer. No justifications are needed in this exercise. (1 point each)

Math 33 AH : Solution to the Final Exam Honors Linear Algebra and Applications 1. True/False: Circle the correct answer. No justifications are needed in this exercise. (1 point each) (1) If A is an invertible

### Recall 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 n-dimensional 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?

### MAT 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

### Section 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,

### MA 242 LINEAR ALGEBRA C1, Solutions to Second Midterm Exam

MA 4 LINEAR ALGEBRA C, Solutions to Second Midterm Exam Prof. Nikola Popovic, November 9, 6, 9:3am - :5am Problem (5 points). Let the matrix A be given by 5 6 5 4 5 (a) Find the inverse A of A, if it exists.

### Solutions to Linear Algebra Practice Problems

Solutions to Linear Algebra Practice Problems. Find all solutions to the following systems of linear equations. (a) x x + x 5 x x x + x + x 5 (b) x + x + x x + x + x x + x + 8x Answer: (a) We create the

### MATH 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 n-dimensional column

### Math 115A HW4 Solutions University of California, Los Angeles. 5 2i 6 + 4i. (5 2i)7i (6 + 4i)( 3 + i) = 35i + 14 ( 22 6i) = 36 + 41i.

Math 5A HW4 Solutions September 5, 202 University of California, Los Angeles Problem 4..3b Calculate the determinant, 5 2i 6 + 4i 3 + i 7i Solution: The textbook s instructions give us, (5 2i)7i (6 + 4i)(

### We seek a factorization of a square matrix A into the product of two matrices which yields an

LU Decompositions We seek a factorization of a square matrix A into the product of two matrices which yields an efficient method for solving the system where A is the coefficient matrix, x is our variable

### Orthogonal Projections and Orthonormal Bases

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) Definition (Orthogonality, length, unit vectors).

### The Second Undergraduate Level Course in Linear Algebra

The Second Undergraduate Level Course in Linear Algebra University of Massachusetts Dartmouth Joint Mathematics Meetings New Orleans, January 7, 11 Outline of Talk Linear Algebra Curriculum Study Group

### SALEM COMMUNITY COLLEGE Carneys Point, New Jersey 08069 COURSE SYLLABUS COVER SHEET. Action Taken (Please Check One) New Course Initiated

SALEM COMMUNITY COLLEGE Carneys Point, New Jersey 08069 COURSE SYLLABUS COVER SHEET Course Title Course Number Department Linear Algebra Mathematics MAT-240 Action Taken (Please Check One) New Course Initiated

### 2.1: MATRIX OPERATIONS

.: MATRIX OPERATIONS What are diagonal entries and the main diagonal of a matrix? What is a diagonal matrix? When are matrices equal? Scalar Multiplication 45 Matrix Addition Theorem (pg 0) Let A, B, and

### Advanced Techniques for Mobile Robotics Compact Course on Linear Algebra. Wolfram Burgard, Cyrill Stachniss, Kai Arras, Maren Bennewitz

Advanced Techniques for Mobile Robotics Compact Course on Linear Algebra Wolfram Burgard, Cyrill Stachniss, Kai Arras, Maren Bennewitz Vectors Arrays of numbers Vectors represent a point in a n dimensional

### University of Ottawa

University of Ottawa Department of Mathematics and Statistics MAT 1302A: Mathematical Methods II Instructor: Alistair Savage Final Exam April 2013 Surname First Name Student # Seat # Instructions: (a)

### Practice Math 110 Final. Instructions: Work all of problems 1 through 5, and work any 5 of problems 10 through 16.

Practice Math 110 Final Instructions: Work all of problems 1 through 5, and work any 5 of problems 10 through 16. 1. Let A = 3 1 1 3 3 2. 6 6 5 a. Use Gauss elimination to reduce A to an upper triangular

### Lecture 5: Singular Value Decomposition SVD (1)

EEM3L1: Numerical and Analytical Techniques Lecture 5: Singular Value Decomposition SVD (1) EE3L1, slide 1, Version 4: 25-Sep-02 Motivation for SVD (1) SVD = Singular Value Decomposition Consider the system

### by the matrix A results in a vector which is a reflection of the given

Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the y-axis We observe that

### 3. 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.

### Name: Section Registered In:

Name: Section Registered In: Math 125 Exam 3 Version 1 April 24, 2006 60 total points possible 1. (5pts) Use Cramer s Rule to solve 3x + 4y = 30 x 2y = 8. Be sure to show enough detail that shows you are

### A Introduction to Matrix Algebra and Principal Components Analysis

A Introduction to Matrix Algebra and Principal Components Analysis Multivariate Methods in Education ERSH 8350 Lecture #2 August 24, 2011 ERSH 8350: Lecture 2 Today s Class An introduction to matrix algebra

### MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix.

MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix. Inverse matrix Definition. Let A be an n n matrix. The inverse of A is an n n matrix, denoted

### Similar matrices and Jordan form

Similar matrices and Jordan form We ve nearly covered the entire heart of linear algebra once we ve finished singular value decompositions we ll have seen all the most central topics. A T A is positive

### Inner 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

### Inverses. Stephen Boyd. EE103 Stanford University. October 27, 2015

Inverses Stephen Boyd EE103 Stanford University October 27, 2015 Outline Left and right inverses Inverse Solving linear equations Examples Pseudo-inverse Left and right inverses 2 Left inverses a number

### MATH10212 Linear Algebra B Homework 7

MATH22 Linear Algebra B Homework 7 Students are strongly advised to acquire a copy of the Textbook: D C Lay, Linear Algebra and its Applications Pearson, 26 (or other editions) Normally, homework assignments

### Summary of week 8 (Lectures 22, 23 and 24)

WEEK 8 Summary of week 8 (Lectures 22, 23 and 24) This week we completed our discussion of Chapter 5 of [VST] Recall that if V and W are inner product spaces then a linear map T : V W is called an isometry

### MAT 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

### Using row reduction to calculate the inverse and the determinant of a square matrix

Using row reduction to calculate the inverse and the determinant of a square matrix Notes for MATH 0290 Honors by Prof. Anna Vainchtein 1 Inverse of a square matrix An n n square matrix A is called invertible

### Orthogonal 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

### Iterative Methods for Computing Eigenvalues and Eigenvectors

The Waterloo Mathematics Review 9 Iterative Methods for Computing Eigenvalues and Eigenvectors Maysum Panju University of Waterloo mhpanju@math.uwaterloo.ca Abstract: We examine some numerical iterative

### Solution. Area(OABC) = Area(OAB) + Area(OBC) = 1 2 det( [ 5 2 1 2. Question 2. Let A = (a) Calculate the nullspace of the matrix A.

Solutions to Math 30 Take-home prelim Question. Find the area of the quadrilateral OABC on the figure below, coordinates given in brackets. [See pp. 60 63 of the book.] y C(, 4) B(, ) A(5, ) O x Area(OABC)

### An Application of Linear Algebra to Image Compression

An Application of Linear Algebra to Image Compression Paul Dostert July 2, 2009 1 / 16 Image Compression There are hundreds of ways to compress images. Some basic ways use singular value decomposition

### Math Practice Problems for Test 1

Math 290 - Practice Problems for Test 1 UNSUBSTANTIATED ANSWERS MAY NOT RECEIVE CREDIT. 3 4 5 1. Let c 1 and c 2 be the columns of A 5 2 and b 1. Show that b Span{c 1, c 2 } by 6 6 6 writing b as a linear

### Row and column operations

Row and column operations It is often very useful to apply row and column operations to a matrix. Let us list what operations we re going to be using. 3 We ll illustrate these using the example matrix

### Cofactor Expansion: Cramer s Rule

Cofactor Expansion: Cramer s Rule MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Introduction Today we will focus on developing: an efficient method for calculating

### 4.1 VECTOR SPACES AND SUBSPACES

4.1 VECTOR SPACES AND SUBSPACES What is a vector space? (pg 229) A vector space is a nonempty set, V, of vectors together with two operations; addition and scalar multiplication which satisfies the following

### MATH 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

### Linear 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

### (January 14, 2009) End k (V ) End k (V/W )

(January 14, 29) [16.1] Let p be the smallest prime dividing the order of a finite group G. Show that a subgroup H of G of index p is necessarily normal. Let G act on cosets gh of H by left multiplication.

### ISOMETRIES 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

### MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set.

MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. Vector space A vector space is a set V equipped with two operations, addition V V (x,y) x + y V and scalar

### Applied Linear Algebra

Applied Linear Algebra OTTO BRETSCHER http://www.prenhall.com/bretscher Chapter 7 Eigenvalues and Eigenvectors Chia-Hui Chang Email: chia@csie.ncu.edu.tw National Central University, Taiwan 7.1 DYNAMICAL

### Lecture 1: Schur s Unitary Triangularization Theorem

Lecture 1: Schur s Unitary Triangularization Theorem This lecture introduces the notion of unitary equivalence and presents Schur s theorem and some of its consequences It roughly corresponds to Sections

### Math 550 Notes. Chapter 7. Jesse Crawford. Department of Mathematics Tarleton State University. Fall 2010

Math 550 Notes Chapter 7 Jesse Crawford Department of Mathematics Tarleton State University Fall 2010 (Tarleton State University) Math 550 Chapter 7 Fall 2010 1 / 34 Outline 1 Self-Adjoint and Normal Operators

### LECTURE 1 I. Inverse matrices We return now to the problem of solving linear equations. Recall that we are trying to find x such that IA = A

LECTURE I. Inverse matrices We return now to the problem of solving linear equations. Recall that we are trying to find such that A = y. Recall: there is a matri I such that for all R n. It follows that

### Solutions 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

### 18.06 Problem Set 4 Solution Due Wednesday, 11 March 2009 at 4 pm in 2-106. Total: 175 points.

806 Problem Set 4 Solution Due Wednesday, March 2009 at 4 pm in 2-06 Total: 75 points Problem : A is an m n matrix of rank r Suppose there are right-hand-sides b for which A x = b has no solution (a) What

### Matrix Algebra 2.3 CHARACTERIZATIONS OF INVERTIBLE MATRICES Pearson Education, Inc.

2 Matrix Algebra 2.3 CHARACTERIZATIONS OF INVERTIBLE MATRICES Theorem 8: Let A be a square matrix. Then the following statements are equivalent. That is, for a given A, the statements are either all true

### ADVANCED LINEAR ALGEBRA FOR ENGINEERS WITH MATLAB. Sohail A. Dianat. Rochester Institute of Technology, New York, U.S.A. Eli S.

ADVANCED LINEAR ALGEBRA FOR ENGINEERS WITH MATLAB Sohail A. Dianat Rochester Institute of Technology, New York, U.S.A. Eli S. Saber Rochester Institute of Technology, New York, U.S.A. (g) CRC Press Taylor

### MATH36001 Background Material 2015

MATH3600 Background Material 205 Matrix Algebra Matrices and Vectors An ordered array of mn elements a ij (i =,, m; j =,, n) written in the form a a 2 a n A = a 2 a 22 a 2n a m a m2 a mn is said to be

### 17. Inner product spaces Definition 17.1. Let V be a real vector space. An inner product on V is a function

17. Inner product spaces Definition 17.1. Let V be a real vector space. An inner product on V is a function, : V V R, which is symmetric, that is u, v = v, u. bilinear, that is linear (in both factors):

### Linear Algebra

. Linear Algebra Midterm Solutions. (pts) Consider a matrix A, andletb rref(a). (a) Is ker (A) necessarily equal to ker (B)? Explain. (b) Is im (A) necessarily equal to im (B)? Explain. (a) Yes. By construction

### Lecture 6. Inverse of Matrix

Lecture 6 Inverse of Matrix Recall that any linear system can be written as a matrix equation In one dimension case, ie, A is 1 1, then can be easily solved as A x b Ax b x b A 1 A b A 1 b provided that

### MATH 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

### Problem 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

### Presentation 3: Eigenvalues and Eigenvectors of a Matrix

Colleen Kirksey, Beth Van Schoyck, Dennis Bowers MATH 280: Problem Solving November 18, 2011 Presentation 3: Eigenvalues and Eigenvectors of a Matrix Order of Presentation: 1. Definitions of Eigenvalues

### 1 Introduction to Matrices

1 Introduction to Matrices In this section, important definitions and results from matrix algebra that are useful in regression analysis are introduced. While all statements below regarding the columns

### Linear Algebra Review Part 2: Ax=b

Linear Algebra Review Part 2: Ax=b Edwin Olson University of Michigan The Three-Day Plan Geometry of Linear Algebra Vectors, matrices, basic operations, lines, planes, homogeneous coordinates, transformations

### NOTES on LINEAR ALGEBRA 1

School of Economics, Management and Statistics University of Bologna Academic Year 205/6 NOTES on LINEAR ALGEBRA for the students of Stats and Maths This is a modified version of the notes by Prof Laura

### 9.3 Advanced Topics in Linear Algebra

548 93 Advanced Topics in Linear Algebra Diagonalization and Jordan s Theorem A system of differential equations x = Ax can be transformed to an uncoupled system y = diag(λ,, λ n y by a change of variables

### We know a formula for and some properties of the determinant. Now we see how the determinant can be used.

Cramer s rule, inverse matrix, and volume We know a formula for and some properties of the determinant. Now we see how the determinant can be used. Formula for A We know: a b d b =. c d ad bc c a Can we

### 1 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

### Chapter 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

### MATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix.

MATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix. Matrices Definition. An m-by-n matrix is a rectangular array of numbers that has m rows and n columns: a 11

### (a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular.

Theorem.7.: (Properties of Triangular Matrices) (a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular. (b) The product

### The Moore-Penrose Inverse and Least Squares

University of Puget Sound MATH 420: Advanced Topics in Linear Algebra The Moore-Penrose Inverse and Least Squares April 16, 2014 Creative Commons License c 2014 Permission is granted to others to copy,

### Definition: A square matrix A is block diagonal if A has the form A 1 O O O A 2 O A =

The question we want to answer now is the following: If A is not similar to a diagonal matrix, then what is the simplest matrix that A is similar to? Before we can provide the answer, we will have to introduce

### Linear Dependence Tests

Linear Dependence Tests The book omits a few key tests for checking the linear dependence of vectors. These short notes discuss these tests, as well as the reasoning behind them. Our first test checks

### 4 MT210 Notebook 4 3. 4.1 Eigenvalues and Eigenvectors... 3. 4.1.1 Definitions; Graphical Illustrations... 3

MT Notebook Fall / prepared by Professor Jenny Baglivo c Copyright 9 by Jenny A. Baglivo. All Rights Reserved. Contents MT Notebook. Eigenvalues and Eigenvectors................................... Definitions;

### Factorization Theorems

Chapter 7 Factorization Theorems This chapter highlights a few of the many factorization theorems for matrices While some factorization results are relatively direct, others are iterative While some factorization

### Linear Algebra Methods for Data Mining

Linear Algebra Methods for Data Mining Saara Hyvönen, Saara.Hyvonen@cs.helsinki.fi Spring 2007 Lecture 3: QR, least squares, linear regression Linear Algebra Methods for Data Mining, Spring 2007, University