Similarity and Diagonalization. Similar Matrices

Size: px
Start display at page:

Download "Similarity and Diagonalization. Similar Matrices"

Transcription

1 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 P AP = B. If A is similar to B, we write A B. Remarks If A B, we can write, equivalently, that A = P BP or AP = P B. If A B, we can write, equivalently, that A = P BP or AP = P B. The matrix P depends on A and B. It is not unique for a given pair of similar matrices A and B. To see this, simply take A = B = I, in which case I I, since P IP = I for any invertible matrix P. Theorem 4.2. Let A, B and C be n n matrices. a. A A. b. If A B, then B A. c. If A B and B C, then A C. This means that is an equivalence relation. The main problem is to find a good representative in each equivalence class. The real meaning of P AP is that this is the matrix of the same linear transformation (given in the standard basis by the matrix A) in a different basis, which consists of the columns of P. This really much better explains why many properties are the same for A and P AP. Theorem Let A and B be n n matrices with A B. Then a. det A = det B. b. A is invertible if and only if B is invertible. c. A and B have the same rank. d. A and B have the same characteristic polynomial. e. A and B have the same eigenvalues.

2 MATH022 Linear Algebra Brief lecture notes 49 Diagonalization Definition. An n n matrix A is diagonalizable if there is a diagonal matrix D such that A is similar to D that is, if there is an invertible matrix P such that P AP = D. Note that the eigenvalues of D are its diagonal elements, and these are the same eigenvalues as for A. Theorem Let A be an n n matrix. Then A is diagonalizable if and only if A has n linearly independent eigenvectors. More precisely, there exists an invertible matrix P and a diagonal matrix D such that P AP = D if and only if the columns of P are n linearly independent eigenvectors of A and the diagonal entries of D are the eigenvalues of A corresponding to the eigenvectors in P in the same order. Theorem If A is an n n matrix with n distinct eigenvalues, then A is diagonalizable....since eigenvectors for distinct eigenvalues are lin. indep. by Th Theorem Let A be an n n matrix and let λ, λ 2,..., λ k be distinct eigenvalues of A. If B i is a basis for the eigenspace E λi, then B = B B 2 B k (i.e., the total collection of basis vectors for all of the eigenspaces) is linearly independent. Lemma If A is an n n matrix, then the geometric multiplicity of each eigenvalue is less than or equal to its algebraic multiplicity. Theorem The Diagonalization Theorem Let A be an n n matrix whose distinct eigenvalues are λ, λ 2,..., λ k. The following statements are equivalent: a. A is diagonalizable. b. The union B of the bases of the eigenspaces of A (as in Theorem 4.24) contains n vectors (which is equivalent to k i= dim E λ i = n). c. The algebraic multiplicity of each eigenvalue equals its geometric multiplicity and all eigenvalues are real numbers this condition is missing in the textbook!.

3 MATH022 Linear Algebra Brief lecture notes 50 In these theorems the eigenvalues are supposed to be real numbers, although for real matrices there may be some complex roots of the characteristic polynomial (in fact, these theorems remain valid for vector spaces and matrices over C then, of course, one does not need the condition that the eigenvalues be all real). Theorem 4.27 and Th actually give a method to decide whether A is diagonalizable, and if yes, to find P such that P AP is diagonal: the columns of P are vectors of bases of the eigenspaces. Example. For A = the characteristic polynomial is 2 2 λ 2 2 det(a λi) = 2 λ λ = ( λ) ( λ) 4( λ) 4( λ) = = (λ 5)(λ + ) 2. Thus, eigenvalues are 5 and x Eigenspace E : (A ( )I) x = 0; x 2 = 0 0 ; x = x 2 x 3, x 3 0 where x 2, x 3 are free var.; E = s t s s, t R ; t a basis of E :, 0. 0 Eigenspace E 5 : (A 5I) x = 0; x x 2 = 0 0 ; solve this x 3 0 system...: x = x 2 = x 3, where x 3 is a free var.; E 5 = t t t R ; t a basis of E 5 :. Together the dimensions add up to 3, so B 5 B is a basis of R 3, so A is diagonalizable Let P = 0 ; then P AP = (Note that is we arrange the eigenvectors in a different order, then the eigenvalues on the diagonal must be arranged accordingly: let Q = 0 ; 0 then Q AQ = ) 0 0 5

4 MATH022 Linear Algebra Brief lecture notes Example. For A = 0 82 the eigenvalues are 3,, and 7. Since they are distinct, the matrix is diagonalizable (To find that P such that P AP = 0 0, one still needs to solve those linear systems (A (λ)i) x = 0...). Example. For A = the eigenvalue is 3 of alg. multiplicity Eigenspace E 3 : x = 0; matrix has rank 2, so dim E 3 =. So A is not digonalizable. 2 Example. Use diagonalization to find A 00 for A =. Eigenvalues 2 {[ 2 2 are... and 3. Eigenspace E 3 : x = 0; x 2 2 = x 2 ; basis. {} [ ]} ] 2 2 Eigenspace E : x = 0; x 2 2 = x 2 ; basis. Let P = ; 0 then P AP = D =. Now, A = P DP 0 3, so A 00 = (P DP ) 00 = 00 0 /2 /2 P DP P DP P DP = P D 00 P = = [ 0 3 ] /2 /2 0 /2 / /2 / = /2 / = [ /2 /2 ] 3 (/2)

5 MATH022 Linear Algebra Brief lecture notes 52 Orthogonality in R n We introduce the dot product of vectors in R n by setting that is, if then u = u v = u T v; u. u n u v = u T v = u u n and v = v. v n v. v n = u v + u 2 v u n v n. The dot product is frequently called scalar product or inner product; we shall use the latter term in a slightly more general context. Notice the following properties of the dot product which can be easily checked directly or immediately follow from the properties of matrix multiplication. They hold for arbitrary vectors u, v, w R n and arbitrary scalar λ. u v = v u (commutativity). u ( v + w) = u v + u w u (λ v) = λ( v u) (The last two properties are referred to as linearity of the dot product.) u u = u u 2 n and therefore u u 0. Moreover, if u u = 0 then u = 0. We define the length (or norm) v of vector v = v. v n by v = v v = v 2 + v v2 n Orthogonal and Orthonormal Sets of Vectors A set of vectors v, v 2,..., v k

6 MATH022 Linear Algebra Brief lecture notes 53 in R n is called an orthogonal set if all pairs of distinct vectors in the set are orthogonal that is, if v i v j = 0 whenever i j for i, j =, 2,..., k The standard basis e, e 2,..., e n in R n is an orthogonal set, as is any subset of it. illustrates, there are many other possibilities. As the first example Example 5. Show that { v, v 2, v 3 } is an orthogonal set in R 3 if v = 2, v 2 = 0, v 3 = Solution We must show that every pair of vectors from this set is orthogonal. This is true, since v v 2 = 2(0) + () + ( )() = 0 v 2 v 3 = 0() + ( ) + ()() = 0 v v 3 = 2() + ( ) + ( )() = 0 Theorem 5.. If v, v 2,..., v k is an orthogonal set of nonzero vectors in R n, then these vectors are linearly independent. Proof If c, c 2,..., c k are scalars such that c v + c 2 v c k v k = 0, then or, equivalently, (c v + c 2 v c k v k ) v i = 0 v i = 0 Since c ( v v i ) + + c i ( v i v i ) + + c k ( v k v i ) = 0 () v, v 2,..., v k is an orthogonal set, all of the dot products in equation () are zero, except v i v i. Thus, equation () reduces to c i ( v i v i ) = 0

7 MATH022 Linear Algebra Brief lecture notes 54 Now, v i v i 0 because v i 0 by hypothesis. So we must have c i = 0. The fact that this is true for all i =,..., k implies that v, v 2,..., v k is a linearly independent set. Remark. Thanks to the Theorem 5., we know that if a set of vectors is orthogonal, it is automatically linearly independent. For example, we can immediately deduce that the three vectors in Example 5. are linearly independent. Contrast this approach with the work needed to establish their linear independence directly! An orthogonal basis for a subspace W of R n is a basis of W that is an orthogonal set. Example 5.2. The vectors v = 2, v 2 = 0, v 3 = from Example 5. are orthogonal and, hence, linearly independent. Since any three linearly independent vectors in R 3 form a basis in R 3, by the Fundamental Theorem of Invertible Matrices, it follows that v, v 2, v 3 is an orthogonal basis for R 3. Theorem 5.2 Let { v, v 2,..., v k } be an orthogonal basis for a subspace W of R n and let w be any vector in W. Then the unique scalars c, c 2,..., c k such that are given by Proof Since w = c v + c 2 v c k v k c i = w v i v i v i for i =,..., k v, v 2,..., v k is a basis for W, we know that there are unique scalars c, c 2,..., c k such that w = c v + c 2 v c k v k (from Theorem 3.29). To establish the formula for c i, we take the dot product of this linear combination with v i to obtain w v i = (c v + c 2 v c k v k ) v i = c ( v v i ) + + c i ( v i v i ) + + c k ( v k v i )

8 MATH022 Linear Algebra Brief lecture notes 55 = c i ( v i v i ) since v j v i = 0 for j i. Since v i 0, v i v i 0. Dividing by v i v i, we obtain the desired result. A unit vector is a vector of unit length. Notice that if v 0 then u = v v is a unit vector collinear (directed along the same line) as v: v = v u. A set of vectors in R n is an orthonormal set if it is an orthogonal set of unit vectors. An orthonormal basis for a subspace W of R n is a basis of W that is an orthonormal set. Theorem 5.3 Let { q, q 2,..., q k } be an orthonormal basis for a subspace W of R n and let w be any vector in W. Then w = ( w q ) q + ( w q 2 ) q ( w q k ) q k and this representation is unique. Theorem 5.4. The columns of an m n matrix Q form an orthonormal set if and only if Q T Q = I n. Proof. We need to show that (Q T Q) ij = { 0 if i j if i = j Let q i denote the ith column of Q (and, hence, the ith row of Q T ). Since the (i, j) entry of Q T Q is the dot product of the ith row of Q T and the jth column of Q, it follows that (Q T Q) ij = q i q j (2) by the definition of matrix multiplication. Now the columns of Q form an orthonormal set if and only if { 0 if i j q i q j = if i = j which, by equation (2) holds if and only if { (Q T 0 if i j Q) ij = if i = j

9 MATH022 Linear Algebra Brief lecture notes 56 This completes the proof. If the matrix Q in Theorem 5.4 is a square matrix, is has a special name. An n n matrix Q whose columns form an orthonormal set is called an orthogonal matrix. The most important fact about orthogonal matrices is given by the next theorem. Theorem 5.5. A square matrix Q is orthogonal if and only if Q = Q T. Proof. By Theorem 5.4, Q is orthogonal if and only if Q T Q = I. This is true if and only if Q is invertible and Q = Q T, by Theorem 3.3. Example Each of the following matrices is orthogonal: 0, 0 0, 0 [ ] / 2 / 2 / 2 /, 2 [ cos α sin α ] sin α cos α Theorem 5.6. equivalent: Let Q be an n n matrix. The following statements are a. Q is orthogonal. b. Q x = x for every x in R n. c. Q x Q y = x y for every x and y in R n. If Q is an orthogonal matrix, then its rows form an or- Theorem 5.7. thonormal set. Theorem 5.8. Let Q be an orthogonal matrix. a. Q is orthogonal. b. det Q = ±. c. If λ is an eigenvalue of Q, then λ =. d. If Q and Q 2 are orthogonal n n matrices, then so is Q Q 2.

Orthogonal Diagonalization of Symmetric Matrices

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

More information

Chapter 6. Orthogonality

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

More information

1 Eigenvalues and Eigenvectors

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

More information

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). 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 information

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.

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

More information

MAT 242 Test 2 SOLUTIONS, FORM T

MAT 242 Test 2 SOLUTIONS, FORM T MAT 242 Test 2 SOLUTIONS, FORM T 5 3 5 3 3 3 3. Let v =, v 5 2 =, v 3 =, and v 5 4 =. 3 3 7 3 a. [ points] The set { v, v 2, v 3, v 4 } is linearly dependent. Find a nontrivial linear combination of these

More information

Diagonalisation. Chapter 3. Introduction. Eigenvalues and eigenvectors. Reading. Definitions

Diagonalisation. Chapter 3. Introduction. Eigenvalues and eigenvectors. Reading. Definitions Chapter 3 Diagonalisation Eigenvalues and eigenvectors, diagonalisation of a matrix, orthogonal diagonalisation fo symmetric matrices Reading As in the previous chapter, there is no specific essential

More information

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

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

More information

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

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

More information

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 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)(

More information

Section 2.1. Section 2.2. Exercise 6: We have to compute the product AB in two ways, where , B =. 2 1 3 5 A =

Section 2.1. Section 2.2. Exercise 6: We have to compute the product AB in two ways, where , B =. 2 1 3 5 A = Section 2.1 Exercise 6: We have to compute the product AB in two ways, where 4 2 A = 3 0 1 3, B =. 2 1 3 5 Solution 1. Let b 1 = (1, 2) and b 2 = (3, 1) be the columns of B. Then Ab 1 = (0, 3, 13) and

More information

Inner Product Spaces and Orthogonality

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,

More information

Linear Algebra Review. Vectors

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

More information

Recall the basic property of the transpose (for any A): v A t Aw = v w, v, w R n.

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?

More information

MATH 240 Fall, Chapter 1: Linear Equations and Matrices

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

More information

University of Lille I PC first year list of exercises n 7. Review

University of Lille I PC first year list of exercises n 7. Review University of Lille I PC first year list of exercises n 7 Review Exercise Solve the following systems in 4 different ways (by substitution, by the Gauss method, by inverting the matrix of coefficients

More information

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

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

More information

Section 6.1 - Inner Products and Norms

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,

More information

MATH 551 - APPLIED MATRIX THEORY

MATH 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

Inner products on R n, and more

Inner products on R n, and more Inner products on R n, and more Peyam Ryan Tabrizian Friday, April 12th, 2013 1 Introduction You might be wondering: Are there inner products on R n that are not the usual dot product x y = x 1 y 1 + +

More information

Notes on Orthogonal and Symmetric Matrices MENU, Winter 2013

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,

More information

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. 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

More information

4: EIGENVALUES, EIGENVECTORS, DIAGONALIZATION

4: EIGENVALUES, EIGENVECTORS, DIAGONALIZATION 4: EIGENVALUES, EIGENVECTORS, DIAGONALIZATION STEVEN HEILMAN Contents 1. Review 1 2. Diagonal Matrices 1 3. Eigenvectors and Eigenvalues 2 4. Characteristic Polynomial 4 5. Diagonalizability 6 6. Appendix:

More information

13 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.

13 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 three-space, we write a vector in terms

More information

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

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

More information

Chapter 17. Orthogonal Matrices and Symmetries of Space

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

More information

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +

More information

Lecture 1: Schur s Unitary Triangularization Theorem

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

More information

MATH10212 Linear Algebra B Homework 7

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

More information

Inner Product Spaces

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

More information

INTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL

INTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL SOLUTIONS OF THEORETICAL EXERCISES selected from INTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL Eighth Edition, Prentice Hall, 2005. Dr. Grigore CĂLUGĂREANU Department of Mathematics

More information

Presentation 3: Eigenvalues and Eigenvectors of a Matrix

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

More information

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. 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

More information

x + y + z = 1 2x + 3y + 4z = 0 5x + 6y + 7z = 3

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

More information

Linear Algebra Notes for Marsden and Tromba Vector Calculus

Linear Algebra Notes for Marsden and Tromba Vector Calculus Linear Algebra Notes for Marsden and Tromba Vector Calculus n-dimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of

More information

Recall that two vectors in are perpendicular or orthogonal provided that their dot

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

More information

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 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

More information

NOTES on LINEAR ALGEBRA 1

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

More information

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

(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.

More information

1 Sets and Set Notation.

1 Sets and Set Notation. LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most

More information

α = u v. In other words, Orthogonal Projection

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 information

Using determinants, it is possible to express the solution to a system of equations whose coefficient matrix is invertible:

Using determinants, it is possible to express the solution to a system of equations whose coefficient matrix is invertible: Cramer s Rule and the Adjugate Using determinants, it is possible to express the solution to a system of equations whose coefficient matrix is invertible: Theorem [Cramer s Rule] If A is an invertible

More information

[1] Diagonal factorization

[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 information

University of Ottawa

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)

More information

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 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

More information

Sec 4.1 Vector Spaces and Subspaces

Sec 4.1 Vector Spaces and Subspaces Sec 4. Vector Spaces and Subspaces Motivation Let S be the set of all solutions to the differential equation y + y =. Let T be the set of all 2 3 matrices with real entries. These two sets share many common

More information

Solutions to Linear Algebra Practice Problems

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

More information

v w is orthogonal to both v and w. the three vectors v, w and v w form a right-handed set of vectors.

v w is orthogonal to both v and w. the three vectors v, w and v w form a right-handed 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 information

Problems for Advanced Linear Algebra Fall 2012

Problems for Advanced Linear Algebra Fall 2012 Problems for Advanced Linear Algebra Fall 2012 Class will be structured around students presenting complete solutions to the problems in this handout. Please only agree to come to the board when you are

More information

Linear Algebra I. Ronald van Luijk, 2012

Linear Algebra I. Ronald van Luijk, 2012 Linear Algebra I Ronald van Luijk, 2012 With many parts from Linear Algebra I by Michael Stoll, 2007 Contents 1. Vector spaces 3 1.1. Examples 3 1.2. Fields 4 1.3. The field of complex numbers. 6 1.4.

More information

WHICH LINEAR-FRACTIONAL TRANSFORMATIONS INDUCE ROTATIONS OF THE SPHERE?

WHICH LINEAR-FRACTIONAL TRANSFORMATIONS INDUCE ROTATIONS OF THE SPHERE? WHICH LINEAR-FRACTIONAL TRANSFORMATIONS INDUCE ROTATIONS OF THE SPHERE? JOEL H. SHAPIRO Abstract. These notes supplement the discussion of linear fractional mappings presented in a beginning graduate course

More information

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 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

More information

Solution based on matrix technique Rewrite. ) = 8x 2 1 4x 1x 2 + 5x x1 2x 2 2x 1 + 5x 2

Solution based on matrix technique Rewrite. ) = 8x 2 1 4x 1x 2 + 5x x1 2x 2 2x 1 + 5x 2 8.2 Quadratic Forms Example 1 Consider the function q(x 1, x 2 ) = 8x 2 1 4x 1x 2 + 5x 2 2 Determine whether q(0, 0) is the global minimum. Solution based on matrix technique Rewrite q( x1 x 2 = x1 ) =

More information

Chapter 20. Vector Spaces and Bases

Chapter 20. Vector Spaces and Bases Chapter 20. Vector Spaces and Bases In this course, we have proceeded step-by-step through low-dimensional Linear Algebra. We have looked at lines, planes, hyperplanes, and have seen that there is no limit

More information

Inner product. Definition of inner product

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

More information

Introduction to Matrix Algebra

Introduction to Matrix Algebra Psychology 7291: Multivariate Statistics (Carey) 8/27/98 Matrix Algebra - 1 Introduction to Matrix Algebra Definitions: A matrix is a collection of numbers ordered by rows and columns. It is customary

More information

NOTES ON LINEAR TRANSFORMATIONS

NOTES 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 information

C 1 x(t) = e ta C = e C n. 2! A2 + t3

C 1 x(t) = e ta C = e C n. 2! A2 + t3 Matrix Exponential Fundamental Matrix Solution Objective: Solve dt A x with an n n constant coefficient matrix A x (t) Here the unknown is the vector function x(t) x n (t) General Solution Formula in Matrix

More information

MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix.

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

More information

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

(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

More information

Linear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices

Linear 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 information

2.5 Elementary Row Operations and the Determinant

2.5 Elementary Row Operations and the Determinant 2.5 Elementary Row Operations and the Determinant Recall: Let A be a 2 2 matrtix : A = a b. The determinant of A, denoted by det(a) c d or A, is the number ad bc. So for example if A = 2 4, det(a) = 2(5)

More information

Applied Linear Algebra I Review page 1

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

More information

some algebra prelim solutions

some algebra prelim solutions some algebra prelim solutions David Morawski August 19, 2012 Problem (Spring 2008, #5). Show that f(x) = x p x + a is irreducible over F p whenever a F p is not zero. Proof. First, note that f(x) has no

More information

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a

More information

Facts About Eigenvalues

Facts About Eigenvalues Facts About Eigenvalues By Dr David Butler Definitions Suppose A is an n n matrix An eigenvalue of A is a number λ such that Av = λv for some nonzero vector v An eigenvector of A is a nonzero vector v

More information

LINEAR ALGEBRA. September 23, 2010

LINEAR ALGEBRA. September 23, 2010 LINEAR ALGEBRA September 3, 00 Contents 0. LU-decomposition.................................... 0. Inverses and Transposes................................. 0.3 Column Spaces and NullSpaces.............................

More information

Eigenvalues and eigenvectors of a matrix

Eigenvalues and eigenvectors of a matrix Eigenvalues and eigenvectors of a matrix Definition: If A is an n n matrix and there exists a real number λ and a non-zero column vector V such that AV = λv then λ is called an eigenvalue of A and V is

More information

MATH 511 ADVANCED LINEAR ALGEBRA SPRING 2006

MATH 511 ADVANCED LINEAR ALGEBRA SPRING 2006 MATH 511 ADVANCED LINEAR ALGEBRA SPRING 26 Sherod Eubanks HOMEWORK 1 1.1 : 3, 5 1.2 : 4 1.3 : 4, 6, 12, 13, 16 1.4 : 1, 5, 8 Section 1.1: The Eigenvalue-Eigenvector Equation Problem 3 Let A M n (R). If

More information

Numerical Analysis Lecture Notes

Numerical Analysis Lecture Notes Numerical Analysis Lecture Notes Peter J. Olver 6. Eigenvalues and Singular Values In this section, we collect together the basic facts about eigenvalues and eigenvectors. From a geometrical viewpoint,

More information

Diagonal, Symmetric and Triangular Matrices

Diagonal, Symmetric and Triangular Matrices Contents 1 Diagonal, Symmetric Triangular Matrices 2 Diagonal Matrices 2.1 Products, Powers Inverses of Diagonal Matrices 2.1.1 Theorem (Powers of Matrices) 2.2 Multiplying Matrices on the Left Right by

More information

Linear Dependence Tests

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

More information

MA 242 LINEAR ALGEBRA C1, Solutions to Second Midterm Exam

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.

More information

Calculus and linear algebra for biomedical engineering Week 4: Inverse matrices and determinants

Calculus and linear algebra for biomedical engineering Week 4: Inverse matrices and determinants Calculus and linear algebra for biomedical engineering Week 4: Inverse matrices and determinants Hartmut Führ fuehr@matha.rwth-aachen.de Lehrstuhl A für Mathematik, RWTH Aachen October 30, 2008 Overview

More information

Matrix Algebra LECTURE 1. Simultaneous Equations Consider a system of m linear equations in n unknowns: y 1 = a 11 x 1 + a 12 x 2 + +a 1n x n,

Matrix Algebra LECTURE 1. Simultaneous Equations Consider a system of m linear equations in n unknowns: y 1 = a 11 x 1 + a 12 x 2 + +a 1n x n, LECTURE 1 Matrix Algebra Simultaneous Equations Consider a system of m linear equations in n unknowns: y 1 a 11 x 1 + a 12 x 2 + +a 1n x n, (1) y 2 a 21 x 1 + a 22 x 2 + +a 2n x n, y m a m1 x 1 +a m2 x

More information

7 - Linear Transformations

7 - Linear Transformations 7 - Linear Transformations Mathematics has as its objects of study sets with various structures. These sets include sets of numbers (such as the integers, rationals, reals, and complexes) whose structure

More information

1 Orthogonal projections and the approximation

1 Orthogonal projections and the approximation Math 1512 Fall 2010 Notes on least squares approximation Given n data points (x 1, y 1 ),..., (x n, y n ), we would like to find the line L, with an equation of the form y = mx + b, which is the best fit

More information

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 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):

More information

Lecture Notes 2: Matrices as Systems of Linear Equations

Lecture 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 information

MATH36001 Background Material 2015

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

More information

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.

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)

More information

Section 4.4 Inner Product Spaces

Section 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 information

Lecture Note on Linear Algebra 15. Dimension and Rank

Lecture Note on Linear Algebra 15. Dimension and Rank Lecture Note on Linear Algebra 15. Dimension and Rank Wei-Shi Zheng, wszheng@ieee.org, 211 November 1, 211 1 What Do You Learn from This Note We still observe the unit vectors we have introduced in Chapter

More information

Notes on Determinant

Notes on Determinant ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 9-18/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without

More information

CHARACTERISTIC ROOTS AND VECTORS

CHARACTERISTIC ROOTS AND VECTORS CHARACTERISTIC ROOTS AND VECTORS 1 DEFINITION OF CHARACTERISTIC ROOTS AND VECTORS 11 Statement of the characteristic root problem Find values of a scalar λ for which there exist vectors x 0 satisfying

More information

5.3 ORTHOGONAL TRANSFORMATIONS AND ORTHOGONAL MATRICES

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

More information

Linear Least Squares

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

More information

Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

More information

ISOMETRIES OF R n KEITH CONRAD

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

More information

2.1: Determinants by Cofactor Expansion. Math 214 Chapter 2 Notes and Homework. Evaluate a Determinant by Expanding by Cofactors

2.1: Determinants by Cofactor Expansion. Math 214 Chapter 2 Notes and Homework. Evaluate a Determinant by Expanding by Cofactors 2.1: Determinants by Cofactor Expansion Math 214 Chapter 2 Notes and Homework Determinants The minor M ij of the entry a ij is the determinant of the submatrix obtained from deleting the i th row and the

More information

T ( a i x i ) = a i T (x i ).

T ( 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 information

LINEAR ALGEBRA W W L CHEN

LINEAR 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 information

Linear Algebra Notes

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

More information

Notes on Jordan Canonical Form

Notes on Jordan Canonical Form Notes on Jordan Canonical Form Eric Klavins University of Washington 8 Jordan blocks and Jordan form A Jordan Block of size m and value λ is a matrix J m (λ) having the value λ repeated along the main

More information

Linear Algebra Test 2 Review by JC McNamara

Linear Algebra Test 2 Review by JC McNamara Linear Algebra Test 2 Review by JC McNamara 2.3 Properties of determinants: det(a T ) = det(a) det(ka) = k n det(a) det(a + B) det(a) + det(b) (In some cases this is true but not always) A is invertible

More information

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

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;

More information

Chapter 8. Matrices II: inverses. 8.1 What is an inverse?

Chapter 8. Matrices II: inverses. 8.1 What is an inverse? Chapter 8 Matrices II: inverses We have learnt how to add subtract and multiply matrices but we have not defined division. The reason is that in general it cannot always be defined. In this chapter, we

More information

1 Spherical Kinematics

1 Spherical Kinematics ME 115(a): Notes on Rotations 1 Spherical Kinematics Motions of a 3-dimensional rigid body where one point of the body remains fixed are termed spherical motions. A spherical displacement is a rigid body

More information

The Hadamard Product

The Hadamard Product The Hadamard Product Elizabeth Million April 12, 2007 1 Introduction and Basic Results As inexperienced mathematicians we may have once thought that the natural definition for matrix multiplication would

More information

Linear Codes. In the V[n,q] setting, the terms word and vector are interchangeable.

Linear Codes. In the V[n,q] setting, the terms word and vector are interchangeable. Linear Codes Linear Codes In the V[n,q] setting, an important class of codes are the linear codes, these codes are the ones whose code words form a sub-vector space of V[n,q]. If the subspace of V[n,q]

More information