# APPENDIX: MATHEMATICAL INDUCTION AND OTHER FORMS OF PROOF

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1 ELEMENTARY LINEAR ALGEBRA WORKBOOK/FOR USE WITH RON LARSON S TEXTBOOK ELEMENTARY LINEAR ALGEBRA CREATED BY SHANNON MARTIN MYERS APPENDIX: MATHEMATICAL INDUCTION AND OTHER FORMS OF PROOF When you are done with your homework you should be able to Use the Principle of Mathematical Induction to prove statements involving a positive integer n Prove by contradiction that a mathematical statement is true Use a counterexample to show that a mathematical statement is false Mathematical Induction Mathematical Induction is a method of mathematical used to establish a given statement for all numbers. It is a form of proof. It is done in steps. The first step, known as the case, is to prove the given statement for the first natural number. The second step, known as the step, is to prove that the given statement for any one natural number the given statement for the next natural number. The Principle of Mathematical Induction Let P n be a statement involving the positive integer n. If 1. P 1 is true, and 2. for every integer k, the truth of implies the truth of then the statement P n must be true for all positive n. CREATED BY SHANNON MARTIN GRACEY 1

2 Example 1: Use mathematical induction to prove the formula for every positive integer n. n n n 4 CREATED BY SHANNON MARTIN GRACEY 2

3 PROOF BY CONTRADICTION In mathematical logic, proof by contradiction is described by the following equivalence: implies if and only if implies. One way to prove that is a true statement is to assume that q is true. If this leads you to a statement that you know is, then you have proved that must be. Example 2: Use proof by contradiction to prove the statement. 1 1 a. If a and b are real numbers and 1 a b, then. a b b. If a is a real number and 0 a 1, then 2 a a. CREATED BY SHANNON MARTIN GRACEY 3

4 USING COUNTEREXAMPLES Example 3: Use a counterexample to show that the statement is false. a. The product of two irrational numbers is irrational. b. If f is a polynomial function and f a f b, then a b. CREATED BY SHANNON MARTIN GRACEY 4

5 Section 1.1: INTRODUCTION TO SYSTEMS OF LINEAR EQUATIONS When you are done with your homework you should be able to Recognize a linear equation in n variables Find a parametric representation of a solution set Determine whether a system of linear equations is consistent of inconsistent Use back-substitution and Gaussian elimination to solve a system of linear equations WARM-UP: Solve the system. a. x 8 y = 3 4y 2 b. 3 x y z = 4 2y 4 z 0 z 1 CREATED BY SHANNON MARTIN GRACEY 5

6 DEFINITION OF A LINEAR EQUATION IN n VARIABLES A linear equation in n variables has the form a, a, a,, an The are numbers, and the term b is a real number. The number a1 is the, and is the leading variable. *Linear equations have no or of variables and no variables involved in functions. Example 1: Give an example of a linear equation in three variables. SOLUTIONS AND SOLUTION SETS A solution of a linear equation in n variables is a of n s, s, s,, sn real numbers the values arranged to satisfy the equation when you substitute into the equation. The set of solutions of a linear equation is called its, and when you have found this set, you have the equation. To describe the entire solution set of a linear equation, use a representation. CREATED BY SHANNON MARTIN GRACEY 6

7 Example 2: Solve the linear equation x1 x x x 5x 1. Example 3: Solve the linear equation SYSTEMS OF LINEAR EQUATIONS IN n VARIABLES A system of linear equations in n variables is a set of m equations, each of which is linear in the same n variables. a x a x a x a x b n n 1 a x a x a x a x b n n 2 a x a x a x a x b n n 3 a x a x a x a x b m1 1 m2 2 m3 3 mn n m SOLUTIONS OF SYSTEMS OF LINEAR EQUATIONS A solution of a system of linear equations is a of s, s, s,, sn numbers that is a solution of each of the linear equations in the CREATED BY SHANNON MARTIN GRACEY 7

8 Example 4: Graph the following linear systems and determine the solution(s), if a solution exists. a. x y 12 x y 2 b. x y 12 x y 2 c. x y 12 2x 2y 24 CREATED BY SHANNON MARTIN GRACEY 8

9 NUMBER OF SOLUTIONS OF A SYSTEM OF EQUATIONS For a system of linear equations, precisely one of the following is true. 1. The system has solution. ( system). 2. The system has solutions ( system) 3. The system has ( system). OPERATIONS THAT PRODUCE EQUIVALENT SYSTEMS Each of the following operations on a system of linear equations produces an system. 1. two equations. 2. an equation by a constant. 3. a of an equation to equation. The idea is to get the system into form. a x a x a x b a x a x b a x b CREATED BY SHANNON MARTIN GRACEY 9

10 Example 5: Solve the system of linear equations. a. 2 1 x1 x x x b. x x x x 3x 2 x x x CREATED BY SHANNON MARTIN GRACEY 10

11 c. 5 x 3x 2x x 4 x x x 11x 4x CREATED BY SHANNON MARTIN GRACEY 11

12 Section 1.2: GAUSSIAN ELIMINATION AND GAUSS-JORDAN ELIMINATION When you are done with your homework you should be able to Determine the size of a matrix and write an augmented or coefficient matrix from a system of linear equations Use matrices and Gaussian elimination with back-substitution to solve a system of linear equations Use matrices and Gauss-Jordan elimination to solve a system of linear equations Solve a homogeneous system of linear equations TYPES OF SOLUTIONS 2 Equations, 2 Variables 3 Equations, 3 Variables CREATED BY SHANNON MARTIN GRACEY 12

13 DEFINITION OF A MATRIX If m and n are positive integers, an m is a array n matrix (read ) matrix A a a a a a a a a a n n m1 m2 mn in which each, a ij, of the matrix is a number. An m n matrix has m rows and n columns. Matrices are usually denoted by letters. *The entry aij is located in the ith row and the jth column. The index i is called the because it identifies the row in which the entry lies, and the index j is called the because it identifies the column in which the entry lies. **A matrix with m rows and n columns is said to be of. When, the matrix is called of order n and the entries a11, a22, a33, are called the entries. CREATED BY SHANNON MARTIN GRACEY 13

14 Example 1: Consider the system of equations we solved in x 3x 2x x 4 x x x 11x 4x a. What is the coefficient matrix? What is the size of the coefficient matrix? b. What is the augmented matrix? ELEMENTARY ROW OPERATIONS 1. two rows. 2. a row by a constant. 3. a of a row to row. 4. ( ) any 2 rows. CREATED BY SHANNON MARTIN GRACEY 14

15 ROW-ECHELON FORM AND REDUCED ROW-ECHELON FORM A matrix in form has the following properties. 1. Any rows consisting entirely of occur at the bottom of the matrix. 2. For each row that does not consist entirely of zeros, the first nonzero entry is (called a leading ). 3. For two successive nonzero rows, the leading 1 in the higher row is farther to the than the leading 1 in the lower row. A matrix in row-echelon form is in form when every column that has a leading 1 has in every position above and below its leading 1. GAUSSIAN ELIMINATION WITH BACK SUBSTITUTION 1. Write the matrix of the system of linear equations. 2. Use elementary row operations to the matrix in rowechelon form. 3. Write the system of linear equations corresponding to the matrix in form, and use back substitution to find the solution. CREATED BY SHANNON MARTIN GRACEY 15

16 Example 2: Solve the system using Gaussian elimination with back substitution. 5 x 3x 2x x 4 x x x 11x 4x CREATED BY SHANNON MARTIN GRACEY 16

17 Example 3: Solve the system using Gauss-Jordan elimination. x + x 5x 3 x x x x x CREATED BY SHANNON MARTIN GRACEY 17

18 HOMOGENEOUS SYSTEMS OF LINEAR EQUATIONS Systems of equations in which each of the terms is zero are called. A homogeneous system of m equations in n variables has the form a x a x a x a x n a x a x a x a x n a x a x a x a x n a x a x a x a x 0 m1 1 m2 2 m3 3 mn n n n n **Homogenous linear systems either have the solution, or solutions Example 4: Solve the homogeneous linear system corresponding to the given coefficient matrix THEOREM 1.1: THE NUMBER OF SOLUTIONS OF A HOMOGENEOUS SYSTEM Every homogeneous system of linear equations is. If the system has fewer equations than variables, then it must have solutions. CREATED BY SHANNON MARTIN GRACEY 18

19 Section 1.3: APPLICATIONS OF SYSTEMS OF LINEAR EQUATIONS When you are done with your homework you should be able to Set up and solve a system of equations to fit a polynomial function to a set of data points Set up and solve a system of equations to represent a network POLYNOMIAL CURVE FITTING Suppose n points in the xy-plane represent a collection of and you are asked to find a function of degree whose graph passes through the specified points. This procedure is called. If all x-coordinates are distinct, then there is precisely polynomial function of degree n 1 (or less) that fits the n points. To solve for the n of p x, each of the n points into the polynomial function and obtain n equations in variables a0, a1, a2,, an 1. a a x a x a x y 2 n n1 1 1 a a x a x a x y 2 n n1 2 2 a a x a x a x y 2 n n1 3 3 a a x a x a x y 2 n1 0 1 n 2 n n1 n n CREATED BY SHANNON MARTIN GRACEY 19

20 Example 1: Determine the polynomial function whose graph passes through the points, and graph the polynomial function, showing the given points. 2,4, 3,4, 4,4 CREATED BY SHANNON MARTIN GRACEY 20

21 Example 2: The table shows the U.S. population figures for the years 1940, 1950, 1960, and (Source: U.S. Census Bureau) Year Population (in millions) a. Find a cubic polynomial that fits these data and use it to estimate the population in b. The actual population in 1980 was 227 million. How does your estimate to compare? CREATED BY SHANNON MARTIN GRACEY 21

22 NETWORK ANALYSIS Networks composed of and are used as models in fields like economics, traffic analysis, and electrical engineering. In a network model you assume that the total into a is equal to the total flow of the junction. CREATED BY SHANNON MARTIN GRACEY 22

23 Example 3: The figure shows the flow of traffic through a network of streets. 400 x x 2 x 3 x a. Solve this system for x i, i 1,2,,5. x 5 CREATED BY SHANNON MARTIN GRACEY 23

24 b. Find the traffic flow when x3 0 and x c. Find the traffic flow when x3 x CREATED BY SHANNON MARTIN GRACEY 24

25 Section 2.1: OPERATIONS WITH MATRICES When you are done with your homework you should be able to Determine whether two matrices are equal Add and subtract matrices and multiply a matrix by a scalar Multiply two matrices Use matrices to solve a system of equations Partition a matrix and write a linear combination of column vectors DEFINITION OF EQUALITY OF MATRICES Two matrices A aij and B bij are when they have the same and for and. Example 1: Are matrices A and B equal? Please explain. A B CREATED BY SHANNON MARTIN GRACEY 25

26 Example 2: Find x and y x y 3 8 A matrix that has only one is called a or. A matrix that has only one is called a or. Boldface lowercase letters often designate and. DEFINITION OF MATRIX ADDITION If A the m aij and B bij n matrix given by are matrices of size m n, then their is The sum of two matrices of different sizes is. CREATED BY SHANNON MARTIN GRACEY 26

27 DEFINITION OF SCALAR MULTIPLICATION If A a is an m n matrix and c is a scalar, then the ij of A by c is the matrix given by You can use to represent the scalar product. If A and B are of the same size, then A B represents the sum of and. Example 3: For the matrices A and B , find a. A B b. 2A B CREATED BY SHANNON MARTIN GRACEY 27

28 DEFINITION OF MATRIX MULTIPLICATION If A a is an m n ij matrix and ij B b is an n p matrix, then the AB is an m p matrix. where To find an entry in the ith row and the jth column of the product AB, multiply the in the row of A by the corresponding entries in the column of B and and then the results. Example 4: Find the product AB, where A 4 3 and B 12 7 CREATED BY SHANNON MARTIN GRACEY 28

29 Example 5: Consider the matrices A and B A and B 6 13 a. Find A B. c. Find B A. b. Find AB. d. Find BA. e. Is matrix addition commutative? f. Is matrix multiplication commutative? CREATED BY SHANNON MARTIN GRACEY 29

30 Example 6: Multiply. a11 a12 a13 x1 a a a x a31 a32 a 33 x 3 SYSTEMS OF LINEAR EQUATIONS The system can be written as or equivalently, CREATED BY SHANNON MARTIN GRACEY 30

31 Example 7: Write the system of equations in the form Ax b and solve this matrix equation for x. 2 x + 3 x 5 x 1 2 4x PARTITIONED MATRICES a a a a a a A a a a n n m1 m2 mn x1 x 2 x xn b b1 b 2 bm CREATED BY SHANNON MARTIN GRACEY 31

32 LINEAR COMBINATIONS The matrix product Ax is a linear combination of the vectors a1, a2, a3,, an that form the matrix A. The system Ax b is consistent if and only if b can be expressed as such a, where the of the linear combination are a of the system. Example 8: Write the column matrix b as a linear combination of the columns of A. 3 5 A b 4 32 CREATED BY SHANNON MARTIN GRACEY 32

33 Example 9: Find the products AB and BA for the diagonal matrices. A B Example 10: Let A and B be matrices such that the product of AB is defined. Show that if A has two identical rows, then the corresponding two rows of AB are also identical. Proof: CREATED BY SHANNON MARTIN GRACEY 33

34 Section 2.2: PROPERTIES OF MATRIX OPERATIONS When you are done with your homework you should be able to Use the properties of matrix addition, scalar multiplication, and zero matrices Use the properties of matrix multiplication and the identity matrix Find the transpose of a matrix THEOREM 2.1: PROPERTIES OF MATRIX ADDITION AND SCALAR MULTIPLICATION If A, B, and C are m n matrices, and c and d are scalars, then the following properties are true. 1. A B Commutative property of addition 2. A B C Associative property of addition CREATED BY SHANNON MARTIN GRACEY 34

35 3. cd A Associative property of multiplication 4. 1A Multiplicative Identity 5. c A B Distributive property 6. c d A Distributive property CREATED BY SHANNON MARTIN GRACEY 35

36 Example 1: For the matrices below, c 2, and d A B C a. c A C b. cdb c. ca B C CREATED BY SHANNON MARTIN GRACEY 36

37 THEOREM 2.2: PROPERTIES OF ZERO MATRICES If A is an m n matrix, and c is a scalar, then the following properties are true. A O 1. mn 2. A A 3. If ca O, then. Example 2: Solve for X in the equation, given 2 1 A and B a. X 3A 2B b. 2A 4B 2X CREATED BY SHANNON MARTIN GRACEY 37

38 THEOREM 2.3: PROPERTIES OF MATRIX MULTIPLICATION If A, B, and C are matrices (with sizes such that the given matrix products are defined), and c is a scalar, then the following properties are true. 1. A BC Associative property of multiplication 2. AB C Distributive property of multiplication 3. A B C Distributive property of multiplication 4. c AB ca B Example 3: Show that AC BC, even though A B A B C CREATED BY SHANNON MARTIN GRACEY 38

39 Example 4: Show that AB O, even though A O and B O. A B THEOREM 2.4: PROPERTIES OF THE IDENTITY MATRIX If A is an m n matrix, then the following properties are true. 1. AIn 2. I A m CREATED BY SHANNON MARTIN GRACEY 39

40 THEOREM 2.5: NUMBER OF SOLUTIONS OF A LINEAR SYSTEM For a system of linear equations, precisely one of the following is true. 1. The system has exactly solution. 2. The system has many solutions. 3. The system has solution. THE TRANSPOSE OF A MATRIX The transpose of a matrix is denoted and is formed by writing its as. Example 5: Find the transpose of the matrix A THEOREM 2.5: PROPERTIES OF TRANSPOSES If A and B are matrices (with sizes such that the given matrix operations are defined), and c is a scalar, then the following properties are true. T 1. A T Transpose of a transpose CREATED BY SHANNON MARTIN GRACEY 40

41 2. A B T Transpose of a sum 3. ca T Transpose of a scalar multiple 4. AB T Transpose of a product CREATED BY SHANNON MARTIN GRACEY 41

42 Example 6: Find a) symmetric. T A A and b) T AA. Show that each of these products is A Example 7: A square matrix is called skew-symmetric when. Prove that if A and B are skew-symmetric matrices, then A B is skew-symmetric. A T A CREATED BY SHANNON MARTIN GRACEY 42

43 Section 2.3: THE INVERSE OF A MATRIX When you are done with your homework you should be able to Find the inverse of a matrix (if it exists) Use properties of inverse matrices Use an inverse matrix to solve a system of linear equations DEFINITION OF THE INVERSE OF A MATRIX An n n matrix A is or when there exists an n n matrix B such that where I n is the matrix of order n. The matrix B is called the ( ) of A. A matrix that does not have an inverse is called noninvertible or. *Nonsquare matrices do not have. Example 1: For the matrices below, show that B is the inverse of A. 1 1 A 1 2 B CREATED BY SHANNON MARTIN GRACEY 43

44 THEOREM 2.7: UNIQUENESS OF AN INVERSE If A is an invertible matrix, then its inverse is unique. The inverse of is denoted. Proof: FINDING THE INVERSE OF A MATRIX BY GAUSS-JORDAN ELIMINATION Let A be a square matrix of order n. 1. Write the matrix that consists of the given matrix A on the left and the n n matrix on the right to obtain. This process is called matrix I to matrix A. 2. If possible, row reduce to using elementary row operations on the entire matrix. The result will be the matrix. If this is not possible, then A is noninvertible (or ). 3. Check your work by multiplying to see that. CREATED BY SHANNON MARTIN GRACEY 44

45 Example 2: Find the inverse of the matrix (if it exists), by solving the matrix equation AX I. A Example 3: Find the inverse of the matrix (if it exists). a. 1 2 A 2 3 CREATED BY SHANNON MARTIN GRACEY 45

46 b A CREATED BY SHANNON MARTIN GRACEY 46

47 THEOREM 2.8: PROPERTIES OF INVERSE MATRICES If A is an invertible matrix, k is a positive integer, and c is a nonzero scalar, then 1 A k T, A, ca, and A are invertible and the following are true A 1 k 2. A 1 3. ca 1 T 4. A 1 THEOREM 2.9: THE INVERSE OF A PRODUCT If A and B are invertible matrices of order n, then AB is invertible and AB B A Proof:. CREATED BY SHANNON MARTIN GRACEY 47

48 Example 4: Use the inverse matrices below for the following problems A B a. AB 1 T b. A 1 c. 7A 1 CREATED BY SHANNON MARTIN GRACEY 48

49 THEOREM 2.10: CANCELLATION PROPERTIES If C is an invertible matrix, then the following properties hold true. 1. If AC BC then A B. Right cancellation property 2. If CA CB then A B. Left cancellation property THEOREM 2.11: SYSTEMS OF EQUATIONS WITH UNIQUE SOLUTIONS If A is an invertible matrix, then the system of linear equations Ax b has a unique solution given by x A 1 b. Proof: CREATED BY SHANNON MARTIN GRACEY 49

50 Example 5: Use an inverse matrix to solve the system of equations. x x 2 x x 2 x x x x x CREATED BY SHANNON MARTIN GRACEY 50

51 Section 2.4: ELEMENTARY MATRICES When you are done with your homework you should be able to Factor a matrix into a product of elementary matrices Find and use an LU-factorization of a matrix to solve a system of linear equations DEFINITION OF AN ELEMENTARY MATRIX An n n matrix is called an matrix when it can be obtained from the matrix by a single elementary operation. Example 1: Identify the matrices that are elementary below, show that B is the inverse of A. 2 1 A 0 2 B C THEOREM 2.12: REPRESENTING ELEMENTARY ROW OPERATIONS Let E be the matrix obtained by performing an elementary row operation on. If that same elementary row operation is performed on an matrix A, then the resulting matrix is given by the product. CREATED BY SHANNON MARTIN GRACEY 51

52 Example 2: Let A, B, and C be A B C Find an elementary matrix E such that EA C. CREATED BY SHANNON MARTIN GRACEY 52

53 Example 3: Find a sequence of elementary matrices that can be used to write the matrix in row-echelon form CREATED BY SHANNON MARTIN GRACEY 53

54 DEFINITION OF ROW EQUIVALENCE Let A and B be m n matrices. Matrix B is to A when there exists a finite number of matrices, such that THEOREM 2.13: ELEMENTARY MATRICES ARE INVERTIBLE If E is an elementary matrix, then matrix. 1 E exists and is an Example 4: Find the inverse of the elementary matrix CREATED BY SHANNON MARTIN GRACEY 54

55 THEOREM 2.15: EQUIVALENT CONDITIONS If A is an n n matrix, then the following statements are equivalent. 1. A is. 2. Ax b has a solution for every column matrix. 3. Ax O has only the solution. 4. A is to. 5. A can be written as the product of matrices. THE LU-FACTORIZATION DEFINITION OF LU-FACTORIZATION If the n n matrix A can be written as the product of a lower triangular matrix L and an upper triangular matrix U, then A LU is an LU-factorization of A. CREATED BY SHANNON MARTIN GRACEY 55

56 Example 5: Solve the linear system Ax b by a. Finding an LU-factorization of the coefficient matrix A. b. Solving the lower triangular system Ly b. c. Solving the upper triangular system Ux y. 2 x x x x x 2 x x x 4 1 CREATED BY SHANNON MARTIN GRACEY 56

57 CREATED BY SHANNON MARTIN GRACEY 57

58 Section 2.5: APPLICATIONS OF MATRIX OPERATIONS When you are done with your homework you should be able to Write and use a stochastic matrix Use matrix multiplication to encode and decode messages STOCHASTIC MATRICES Many types of applications involve a finite set of of a given population. The that a member of a population will change from the state to the state is represented by a number, where. A probability of means that the member is certain to change from the jth state to the ith state whereas a probability of means that the member is to change from the jth state to the ith state. P P P P P P P P P P n n n1 n2 nn P is called the of probabilities. At each transition, each member in a given state must either stay in that state or change to another state. Therefore, the sum of the entries in any is. This type of matrix is called. An matrix P is a stochastic matrix when each entry is a number between and inclusive. CREATED BY SHANNON MARTIN GRACEY 58

59 Example 1: Determine whether the matrix is stochastic. A B Example 2: A medical researcher is studying the spread of a virus in a population of 1000 laboratory mice. During any week, there is an 80% probability that an infected mouse will overcome the virus, and during the same week, there is a 10% probability that a noninfected mouse will become infected. One hundred mice are currently infected with the virus. How many will be infected (a) next week and (b) in two weeks? CREATED BY SHANNON MARTIN GRACEY 59

60 CRYPTOGRAPHY A is a message written according to a secret code. Suppose we assign a number to each letter in the alphabet. 0 _ 14 N 1 A 15 O Example 3: Write the uncoded row matrices of size 1 x 3 for the message TARGET IS HOME. 2 B 16 P 3 C 17 Q 4 D 18 R 5 E 19 S 6 F 20 T 7 G 21 U 8 H 22 V 9 I 23 W Example 4: Use the following invertible matrix to encode the message TARGET IS HOME A J 24 X 11 K 25 Y 12 L 26 Z 13 M CREATED BY SHANNON MARTIN GRACEY 60

61 CREATED BY SHANNON MARTIN GRACEY 61

62 Section 3.1: THE DETERMINANT OF A MATRIX When you are done with your homework you should be able to Find the determinant of a 2 x 2 matrix Find the minors and cofactors of a matrix Use expansion by cofactors to find the determinant of a matrix Find the determinant of a triangular matrix Every matrix can be associated with a real number called its. Historically, the use of determinants arose from the recognition of special that occur in the of systems of linear equations. Consider the system a x a x b a x a x b CREATED BY SHANNON MARTIN GRACEY 62

63 So, we have found that DEFINITION OF THE DETERMINANT OF A 2 x 2 MATRIX The of the matrix a A a a a is given by det A. **Note: In this text, and are used interchangeably to represent the determinant of a matrix. In this context, the vertical bars are used to represent the of a matrix as opposed to the value. Example 1: Find A and B. 1 4 A 11 7 B CREATED BY SHANNON MARTIN GRACEY 63

64 DEFINITION OF MINORS AND COFACTORS OF A MATRIX If A is a matrix, then the of the element is the determinant of the matrix obtained by deleting the row and the column of A.The is given by Cij. Example 2: Find the minor and cofactor of a 12.and b 23. a a a A a a a a a a B CREATED BY SHANNON MARTIN GRACEY 64

65 Consider: a a a A a a a a a a B b b b b b b b b b b b b b b b b What do you notice? THEOREM 3.1: EXPANSION BY COFACTORS If A be a square matrix of order n. Then the determinant of A is given by det det A A a C n j1 n A A a C i1 ij ij ij ij (ith row expansion) (jth colomn expansion) Example 3: Confirm that, for 2x2 matrices, this definition yields A a a a a CREATED BY SHANNON MARTIN GRACEY 65

66 Example 4: Find B. B Alternative Method to evaluate the determinant of a 3x3 matrix: Copy the first and second columns of the matrix to form fourth and fifth columns. Then obtain the determinant by adding (or subtracting) the products of the six diagonals. Example 4: Find B. B CREATED BY SHANNON MARTIN GRACEY 66

67 Example 5: Find det A. A What did you notice? THEOREM 3.2: DETERMINANT OF A TRIANGULAR MATRIX If A is a triangular matrix of order n, then its determinant is the of the on the. That is, det A A. CREATED BY SHANNON MARTIN GRACEY 67

68 Example 6: Find the values of, for which the determinant is zero CREATED BY SHANNON MARTIN GRACEY 68

69 Section 3.2: DETERMINANTS AND ELEMENTARY OPERATIONS When you are done with your homework you should be able to Use elementary row operations to evaluate a determinant Use elementary column operations to evaluate a determinant Recognize conditions that yield zero determinants Consider the following two determinants: A B Find the determinant of each matrix. What did you find out? Take a closer look at the two matrices. Do you notice anything? CREATED BY SHANNON MARTIN GRACEY 69

70 THEOREM 3.3: ELEMENTARY ROW OPERATIONS AND DETERMINANTS Let A and B be square matrices. 1. When B is obtained from A by two of A,. 2. When B is obtained from A by a of a row A to another row of A,. 3. When B is obtained from A by a row of A by a c,. NOTE: Theorem 3.3 remains valid when the word column replaces the word row. Operations performed on columns are called elementary column operations. Example 1: Determine which property of determinants the equation illustrates. a. b CREATED BY SHANNON MARTIN GRACEY 70

71 Example 2: Use elementary row or column operations to find the determinant THEOREM 3.4: CONDITIONS THAT YIELD A ZERO DETERMINANT If A is a square matrix, and any one of the following conditions is true, then det A An entire (or ) consists of. 2. Two (or ) are. 3. One (or ) is a of another (or ). CREATED BY SHANNON MARTIN GRACEY 71

72 Example 3: Prove the property. 1 a b 1 abc 1, a 0, b 0, c 0 a b c c. CREATED BY SHANNON MARTIN GRACEY 72

73 Section 3.3: PROPERTIES OF DETERMINANTS When you are done with your homework you should be able to Find the determinant of a matrix product and a scalar multiple of a matrix Find the determinant of an inverse matrix and recognize equivalent conditions for a nonsingular matrix Find the determinant of the transpose of a matrix Example 1: Find A, B, A B, A B, A B and AB. A B CREATED BY SHANNON MARTIN GRACEY 73

74 THEOREM 3.5: DETERMINANT OF A MATRIX PRODUCT If A and B are square matrices of order n, then Example 2: Find 3A and 3B. 1 1 A 3 10 B THEOREM 3.6: DETERMINANT OF A SCALAR MULTIPLE OF A MATRIX If A is a square matrix of order n and c is a scalar, then the determinant of ca is Proof: CREATED BY SHANNON MARTIN GRACEY 74

75 Example 3: Find 1 A 1, A, B, and B. 3 6 A 2 4 B THEOREM 3.7: DETERMINANT OF AN INVERTIBLE MATRIX A square matrix A is invertible (nonsingular) if and only if CREATED BY SHANNON MARTIN GRACEY 75

76 Example 4: Find 1 A and A. 3 3 A 2 1 THEOREM 3.8: DETERMINANT OF AN INVERSE MATRIX If A is an n n invertible matrix, then Proof: CREATED BY SHANNON MARTIN GRACEY 76

77 EQUIVALENT CONDITIONS FOR A NONSINGULAR MATRIX If A is an n n nonsingular matrix, then the following statements are equivalent. 1. A is. 2. Ax b has a solution for every column matrix. 3. Ax 0 has only the solution. 4. A is to. 5. A can be written as the product of matrices. 6.. Example 5: Determine if the system of linear equations has a unique solution. x x x x x x x 2x 2x Example 6: Find T A and A. A THEOREM 3.9: DETERMINANT OF A TRANSPOSE If A is a square matrix, then CREATED BY SHANNON MARTIN GRACEY 77

78 Section 3.4: APPLICATIONS OF DETERMINANTS When you are done with your homework you should be able to Use Cramer s Rule to solve a system of n linear equations Use determinants to find area, volume, and the equations of lines and planes THEOREM 3.11: CRAMER S RULE If a system of n linear equations in n variables has a coefficient matrix A with a nonzero determinant A, then the solution of the system is Where the ith column of Ai is the column of constants in the system of equations. Example 1: If possible, use Cramer s Rule to solve the system. a. x 2x x 4x CREATED BY SHANNON MARTIN GRACEY 78

79 b. 8x 7x 10x x 3 x 5 x x 9 x 2 x CREATED BY SHANNON MARTIN GRACEY 79

80 AREA OF A TRIANGLE IN THE xy-plane The area of a triangle with vertices,,,, and, x y x y x y is where the sign ( ) is chosen to give positive area. Proof: CREATED BY SHANNON MARTIN GRACEY 80

81 Example 2: Find the area of the triangle whose vertices are 1, 1 0, 2., 3, 5, and TEST FOR COLLINEAR POINTS IN THE xy-plane Three points,,,, and, x y x y x y are collinear if and only if TWO-POINT FORM OF THE EQUATION OF A LINE An equation of the line passing through the distinct points given by x y and, 1, 1 x y is 2 2 CREATED BY SHANNON MARTIN GRACEY 81

82 Example 3: Find an equation of the line passing through the points 4,7 and 2, 4. VOLUME OF A TETRAHEDRON The volume of a tetrahedron with vertices x4, y4, z 4 is x y z, x, y, z,,, 1, 1, x y z, and where the sign ( ) is chosen to give positive volume. Example 4: Find the volume of the tetrahedron with vertices 2,1, 1, and 1,1,2. 1,1,1, 0,0,0, CREATED BY SHANNON MARTIN GRACEY 82

83 TEST FOR COPLANAR POINTS IN SPACE Four points, only if x y z, x, y, z, x, y, z, and,, 1, 1, x y z are coplanar if and where the sign ( ) is chosen to give positive volume. THREE-POINT FORM OF THE EQUATION OF A LINE An equation of the plane passing through the distinct points x1, y1, z 1, x y z,and,, 2, 2, 2 x y z is given by CREATED BY SHANNON MARTIN GRACEY 83

84 Section 4.1: VECTORS IN n R When you are done with your homework you should be able to Represent a vector as a directed line segment 2 Perform basic vector operations in R and represent them graphically n Perform basic vector operations in R Write a vector as a linear combination of other vectors VECTORS IN THE PLANE A vector is characterized by two quantities, and, and is represented by a. Geometrically, a in the is represented by a directed line segment with its at the origin and its point at. The same used to represent its terminal point also represents the. That is,. The coordinates x 1 and x 2 are called the of the vector x. Two vectors in the plane u u, u and v, v 1 2 v are 1 2 if and only if and. CREATED BY SHANNON MARTIN GRACEY 84

85 Example 1: Use a directed line segment to represent the vector, and give the graphical representation of the vector operations. a. u 3, 2 b. v 6,1 c. u v d. 2u CREATED BY SHANNON MARTIN GRACEY 85

86 THEOREM 4.1: PROPERTIES OF VECTOR ADDITION AND SCALAR MULTIPLICATION IN THE PLANE Let u, v, and w be vectors in the plane, and let c and d be scalars. 1. u v is a in the plane. under addition 2. u v property of addition Proof: 3. u v w property of addition 4. u 0 additive property 5. u u additive property 6. cu is a in the plane. under scalar mult. Proof: 7. cu v property CREATED BY SHANNON MARTIN GRACEY 86

87 8. c d u property 9. cd 10. u property of multiplication 1 u property NOTE: These ten properties also hold true for n R. Example 2: Let u 1,1,4, v 0,3, 3, and w 7,5, 1 a. u 6v b. v 2 u w IMPORTANT VECTOR SPACES = = the set of. = = the set of all of real numbers. = = the set of all of real numbers. = = the set of all of real numbers. CREATED BY SHANNON MARTIN GRACEY 87

88 DEFINITION OF VECTOR ADDITION AND SCALAR MULTIPLICATION Let and be vectors in, and let. Then the sum of and is defined as the,, and the multiplication of by is defined as the,. Example 3: Let u 0,4,3, 4,4 and 6,8, 3,3, 5 a. v. u v b. 4u 3v THEOREM 4.3: PROPERTIES OF ADDITIVE IDENTITY AND ADDITIVE INVERSE Let v be a vector in true. n R, and let c be a scalar. Then the following properties are 1. The is. Proof: CREATED BY SHANNON MARTIN GRACEY 88

89 2. The is v 4. c0 5. If cv 0, then or. 6. v Example 4: Solve for w, where u 2, 1,3, 4 and 1,8,0,3 a. w u v v. b. w 3v 2u CREATED BY SHANNON MARTIN GRACEY 89

90 LINEAR COMBINATIONS OF VECTORS An important type of problem in linear algebra involves writing one vector as the of of other vectors v1, v2,, vn. The vector, is called a of the vectors 1 2 v, v,, vn. Example 5: If possible, write v as a linear combination of u and w, where 1, 2 1, 1 u and w. a. v 1, 1 b. v 0,3 Example 6: If possible, write v as a linear combination of u 1, u 2, and u 3. v 1,7,2, u, u, and 1 1,3, 5 2 2, 1,3 u. 3 3, 2, 4 CREATED BY SHANNON MARTIN GRACEY 90

91 Section 4.2: VECTOR SPACES When you are done with your homework you should be able to Define a vector space and recognize some important vector spaces Show that a given set is not a vector space VECTOR SPACE A vector space consists of entities: a of, a set of, and operations. When you refer to a vector space, be sure that all four entities are clearly stated or understood. Unless stated otherwise, assume that the set of scalars is the set of numbers. IMPORTANT VECTOR SPACES CONTINUED = the set of all defined on the real line. = the set of all defined on a. = the set of all. = the set of all of degree. = the set of all matrices. = the set of all matrices. CREATED BY SHANNON MARTIN GRACEY 91

92 Example 1: Describe the zero vector (the additive identity) of the vector space. a. C, b. M1,4 Example 2: Describe the additive inverse of a vector in the vector space. a. C, (the set of all realvalued continuous functions defined on the entire real line. b. M1,4 CREATED BY SHANNON MARTIN GRACEY 92

93 DEFINITION OF A VECTOR SPACE Let V be a set on which two operations (vector addition and scalar multiplication) are defined. If the listed axioms are satisfied for every u, v, and w in V and every scalar (real number) c and d, then V is called a vector space. Addition 1. u v is in V. under addition 2. u v property 3. u v w property 4. V has a vector such that for every in V,. additive 5. For every in V, there is a vector in V denoted by such that. additive Scalar Multiplication 6. cu is in. under scalar mult. 7. cu v property 8. c d u property 9. cd 10. u property 1 u identity CREATED BY SHANNON MARTIN GRACEY 93

94 THEOREM 4.4: PROPERTIES OF SCALAR MULTIPLICATION Let v be any element of a vector space V, and let c be any scalar. Then the following properties are true v 2. c0 3. If, then or v Example 3: Determine whether the set, together with the indicated operations, is a vector space. If it is not, then identify at least one of the ten vector space axioms that fails. a. The set of all 2 x 2 matrices of the form a c b 1 CREATED BY SHANNON MARTIN GRACEY 94

95 b. The set 1 x, x : x 2. CREATED BY SHANNON MARTIN GRACEY 95

96 CREATED BY SHANNON MARTIN GRACEY 96

97 c. The set of all 2 x 2 nonsingular matrices with the standard operations. CREATED BY SHANNON MARTIN GRACEY 97

98 Example 4: Rather than use the standard definitions of addition and scalar multiplication in R 3, suppose these two operations are defined as stated 3 below. With these new definitions, is R a vector space? a. x1, y1, z1 x2, y2, z2 x1 x2, y1 y2, z1 z2 c x, y, z cx, cy,0 CREATED BY SHANNON MARTIN GRACEY 98

99 Section 4.3: SUBSPACES OF VECTOR SPACES When you are done with your homework you should be able to Determine whether a subset W of a vector space V is a subspace of V n Determine subspaces of R SUBSPACES In many applications of linear algebra, vector spaces occur as a of larger spaces. A subset of a vector is a when it is a vector with the operations defined in the vector space. Consider the following: W 0, y and V 2 R. DEFINITION OF A SUBSPACE OF A VECTOR SPACE A nonempty subset W of a vector space V is called a of V when is a vector space under the operations of and defined in V. CREATED BY SHANNON MARTIN GRACEY 99

100 THEOREM 4.5: TEST FOR A SUBSPACE If W is a nonempty subset of a vector space V, then W is a subspace of V if and and only if the following closure conditions hold. 1. If u and v are in W, then is in W. 2. If u is in W and c is any scalar, then is in W. Example 1: Verify that W is a subspace of V. W x, y,2x 3 y : x and y a. V R 3 CREATED BY SHANNON MARTIN GRACEY 100

101 b. W is the set of all functions that are differentiable on 1,1. V is the set of all functions that are continuous on 1,1. Example 2: Verify that W is not a subspace of the vector space by giving a specific example that violates the test for vector subspace. a. W is the set of all linear functions ax b, 0 a in, C. b. W is the set of all matrices in M 3,1, of the form a 0 a. T CREATED BY SHANNON MARTIN GRACEY 101

102 THEOREM 4.6: THE INTERSECTION OF TWO SUBSPACES IS A SUBSPACE If V and W are both subspaces of a vector space U, then the intersection of V and W, denoted by, is also a subspace of U. Example 3: Determine whether the subset C, is a subspace of, a. The set of all negative functions: f x 0. C. b. The set of all odd functions: f x f x. CREATED BY SHANNON MARTIN GRACEY 102

103 Example 4: Determine whether the subset of M n, n is a subspace of M, standard operations of matrix addition and scalar multiplication. a. The set of all n x n diagonal matrices. n n with the b. The set of all n x n matrices whose trace is nonzero. Example 5: Determine whether the set W is a subspace of operations. Justify your answer. W x, x, 4 : x and x R with the standard CREATED BY SHANNON MARTIN GRACEY 103

104 Section 4.4: SPANNING SETS AND LINEAR INDEPENDENCE When you are done with your homework you should be able to Write a linear combination of a set of vectors in a vector space V Determine whether a set S of vectors in a vector space V is a spanning set of V Determine whether a set of vectors in a vector space V is linearly independent DEFINITION OF LINEAR COMBINATION OF VECTORS IN A VECTOR SPACE A vector v in a vector space V is called a combination of the vectors u1, u 2,..., u k in V when v can be written in the form where c,,... 1 c c are scalars. c k Example 1: If possible, write each vector as a linear combination of the vectors in S. 1, 2, 2, 2, 1,1 S a. z 4, 3,3 CREATED BY SHANNON MARTIN GRACEY 104

105 S 1, 2, 2, 2, 1,1 b. u 1,1, 1 Example 2: For the matrices A 4 1 and B 1 2 in M 2,2, determine whether the given matrix is a linear combination of A and B CREATED BY SHANNON MARTIN GRACEY 105

106 DEFINITION OF A SPANNING SET OF A VECTOR SPACE S v1, v2,..., vk be a subset of a vector space V. The set S is called a Let set of V when vector in V can be written as a of vectors in S. CREATED BY SHANNON MARTIN GRACEY 106

107 DEFINITION OF THE SPAN OF A SET S v1, v2,..., vk is a set of vectors in a vector space V, then the If of S is the set of all combinations of the vectors in S. The span of S is denoted by When, it is said that V is by, or that spans. THEOREM 4.7: Span(S) IS A SUBSPACE OF V If S v1, v 2,..., v kis a set of a vectors in a vector space V, then span S is a subspace of V. Moreover, span S is the subspace of V that contains S, in the sense that every other subspace of V that contains S must contain span S. Proof: CREATED BY SHANNON MARTIN GRACEY 107

108 2 Example 3: Determine whether the set S spans R. If the set does not span then give a geometric description of the subspace that it does span. a. S 1, 1, 2,1 2 R, 1 1,2, 2, 4,,1 2 b. S CREATED BY SHANNON MARTIN GRACEY 108

109 c. S 1,2, 2, 1, 1,1 DEFINITION OF LINEAR DEPENDENCE AND LINEAR INDEPENDENCE S v1, v2,..., vk in a vector space V is called linearly A set of vector when the vector equation has only the solution If there are also solutions, then S is called linearly. CREATED BY SHANNON MARTIN GRACEY 109

110 TESTING FOR LINEAR INDEPENDENCE AND LINEAR DEPENDENCE S v1, v2,..., v k be a set of vectors in a vector space V. To determine Let whether S is linearly independent of linearly dependent, use the following steps. 1. From the vector equation, write a of linear equations in the variables c1, c2,..., and c k. 2. Use Gaussian elimination to determine whether the system has a solution. 3. If the system has only the solution, c1 c2 c k 0, 0,..., 0, then the set S is linearly independent. If the system has solutions, then S is linearly dependent. Example 4: Determine whether the set S is linearly independent or linearly dependent. a. S 3, 6, 1,2 CREATED BY SHANNON MARTIN GRACEY 110

111 b. S 6,2,1, 1,3,2 c. S 0,0,0,1, 0,0,1,1, 0,1,1,1, 1,1,1,1 Example 5: Determine whether the set of vectors in P2 is linearly independent or linearly dependent. 2 2 S x, x 1 CREATED BY SHANNON MARTIN GRACEY 111

112 Example 6: Determine whether the set of vectors in M 2,2 linearly dependent. A , 4 1 B 0 5, 8 3 C 6 17 is linearly independent or THEOREM 4.8: A PROPERTY OF LINEARLY DEPENDENT SETS S v1, v 2,..., v k, k 2, is linearly dependent if and only if at least one of the vectors v can be written as a linear combination of the other vectors in S. A set Proof: j CREATED BY SHANNON MARTIN GRACEY 112

113 THEOREM 4.8: COROLLARY Two vectors u and v in a vector space V are linearly dependent if and only if one is a of the other. CREATED BY SHANNON MARTIN GRACEY 113

114 Example 7: Show that the set is linearly dependent by finding a nontrivial linear combination of vectors in the set whose sum is the zero vector. Then express one of the vectors in the set as a linear combination of the other vectors in the set. 2,4, 1, 2, 0,6 S CREATED BY SHANNON MARTIN GRACEY 114

115 Section 4.5: BASIS AND DIMENSION When you are done with your homework you should be able to Recognize bases in the vector spaces Find the dimension of a vector space n R, P n, and M, m n DEFINITION OF BASIS S v1, v 2,..., v n in a vector space V is called a A set of vectors for when the following conditions are true. 1. S V. 2. S is linearly. The Standard Basis for S 3 R 1, 0, 0, 0,1, 0, 0, 0,1 CREATED BY SHANNON MARTIN GRACEY 115

116 Example 1: Write the standard basis for the vector space. a. 5 R b. M 3,2 c. P 2 Example 2: Determine whether S is a basis for the indicated vector space. a. 2,1, 0, 0, 1,1 S for 3 R CREATED BY SHANNON MARTIN GRACEY 116

117 b. S 4 t t 2,5 t 3,3t 5, 2t 3 3t 2 for P 3 CREATED BY SHANNON MARTIN GRACEY 117

118 THEOREM 4.9: UNIQUENESS OF BASIS REPRESENTATION S v1, v2,..., vn is a basis for a vector space V, then every vector in V can If be written in one and only one way as a linear combination of vectors in S. Proof: THEOREM 4.10: BASES AND LINEAR DEPENDENCE S v1, v2,..., vn is a basis for a vector space V, then every set containing If more than vectors in V is linearly. CREATED BY SHANNON MARTIN GRACEY 118

119 THEOREM 4.11: NUMBER OF VECTORS IN A BASIS If a vector space V has one basis with, then every basis for V has vectors. Proof: DEFINITION OF DIMENSION OF A VECTOR SPACE If a vector space V has a consisting of vectors, then the number is called the of V, denoted by. When V consists of the vector alone, the dimension of V is defined as. Example 3: Determine the dimension of the vector space. a. 5 R b. M 5,2 c. P 2 CREATED BY SHANNON MARTIN GRACEY 119

120 THEOREM 4.12: BASIS TESTS IN AN n-dimensional SPACE Let V be a vector space of dimension n. 1. If is a linearly independent set of vectors in V, then is a for. 2. If V, then is a for. Example 4: Determine whether S is a basis for the indicated vector space. 1, 2, 1, 1 S for 2 R. CREATED BY SHANNON MARTIN GRACEY 120

121 Example 5: Find a basis for the vector space of all 3 x 3 symmetric matrices. What is the dimension of this vector space? CREATED BY SHANNON MARTIN GRACEY 121

122 Section 4.6: RANK OF A MATRIX AND SYSTEMS OF LINEAR EQUATIONS When you are done with your homework you should be able to Find a basis for the row space, a basis for the column space, and the rank of a matrix Find the nullspace of a matrix Find the solution of a consistent system Ax b in the form x p x h a a A am 1 a 11 1n mn a a A am 1 a 11 1n mn Example 1: Consider the following matrix A a. The row vectors of A are: b. The column vectors of A are: CREATED BY SHANNON MARTIN GRACEY 122

123 DEFINITION OF ROW SPACE AND COLUMN SPACE OF A MATRIX Let A be an m n matrix. 1. The space of A is the of by the vectors of A. n R 2. The space of A is the subspace of by the vectors of A. n R Recall that two matrices are row-equivalent when one can be obtained from the other by operations. THEOREM 4.13: ROW-EQUIVALENT MATRICES HAVE THE SAME ROW SPACE If an m n of A is equal to the row space of B. Proof: matrix A is row-equivalent to an m n matrix B, then the row space CREATED BY SHANNON MARTIN GRACEY 123

124 THEOREM 4.14: BASIS FOR THE ROW SPACE OF A MATRIX If a matrix A is row-equivalent to a matrix B in row-echelon form, then the nonzero row vectors of B form a for the row space of A. Example 2: Find a basis for the row space and column space of the following matrix: A Example 3: Find a basis for the row space and the column space of the following matrix: A CREATED BY SHANNON MARTIN GRACEY 124

125 THEOREM 4.15: ROW AND COLUMN SPACES HAVE EQUAL DIMENSIONS If A is an m n matrix, then the row space and the column space of A have the same. DEFINITION OF THE RANK OF A MATRIX The of the (or ) space of a matrix A is called the of A and is denoted by. Example 4: Find the rank of the matrix from a. example 2, and b. example 3 THEOREM 4.16: SOLUTIONS OF A HOMOGENEOUS SYSTEM If A is an m n matrix, then the set of all solutions of the homogeneous system of linear equations is a of called the of and is denoted. So, The of the nullspace of A is called the of. CREATED BY SHANNON MARTIN GRACEY 125

126 Proof: CREATED BY SHANNON MARTIN GRACEY 126

127 Example 5: Find the nullspace of the following matrix A, and determine the nullity of A. A CREATED BY SHANNON MARTIN GRACEY 127

128 THEOREM 4.17: DIMENSION OF THE SOLUTION SPACE If A is an m n matrix of rank, then the of the solution space of is. That is, Example 6: consider the following homogeneous system of linear equations: x y 0 x y 0 a. Find a basis for the solution space. b. Find the dimension of the solution space. CREATED BY SHANNON MARTIN GRACEY 128

129 THEOREM 4.18: SOLUTIONS OF A NONHOMOGENEOUS LINEAR SYSTEM If x is a particular solution of the nonhomogeneous system Ax p b, then every solution of this system can be written in the form where x h is a solution of the corresponding homogeneous system. Proof: THEOREM 4.19: SOLUTIONS OF A SYSTEM OF LINEAR EQUATIONS The system is consistent if and only if is in the column space of. Proof: CREATED BY SHANNON MARTIN GRACEY 129

130 Example 7: consider the following nonhomogeneous system of linear equations: 2x 4 y 5 z 8 7x 14 y 4z 28 3 x 6 y z 12 a. Determine whether Ax b is consistent. b. If the system is consistent, write the solution in the form x x p x h, where x p is a particular solution of Ax b and x h is a solution of Ax 0. CREATED BY SHANNON MARTIN GRACEY 130

131 Section 4.7: COORDINATES AND CHANGE OF BASIS When you are done with your homework you should be able to Find a coordinate matrix relative to a basis in Find the transition matrix from the basis B to the basis B in Represent coordinates in general n-dimensional spaces n R n R COORDINATE REPRESENTATION RELATIVE TO A BASIS Let B v1, v 2,..., v n be an ordered basis for a vector space V, and let x be a vector in V such that The scalars c1, c2,..., c n are called the of relative to the. The matrix (or coordinate ) of relative to is the matrix in whose are the coordinates of. Note: In, column notation is used for the coordinate matrix. For the vector, the are the coordinates of (relative to the for. So you have CREATED BY SHANNON MARTIN GRACEY 131

132 Example 1: Find the coordinate matrix of x in x 1, 3,0 n R relative to the standard basis. Example 2: Given the coordinate matrix of x relative to a (nonstandard) basis B n for R, find the coordinate matrix of x relative to the standard basis. x B 4,0,7,3, 0,5, 1, 1, 3, 4,2,1, 0,1,5,0 B CREATED BY SHANNON MARTIN GRACEY 132

133 Example 3: Find coordinate matrix of x in 6, 7, 4, 3, x 26,32 B n R relative to the basis B. The last two examples used the procedure called a of. You were given the coordinates of a vector relative to a and were asked to find the coordinates to another basis. The matrix is called the from to, where is the coordinate matrix of relative to, and is the coordinate matrix of relative to. Multiplication by the transition matrix changes a coordinate matrix relative to into a coordinate matrix relative to. CREATED BY SHANNON MARTIN GRACEY 133

134 Change of basis from to : Change of basis from to : The change of basis problem in example 3 can be represented by the matrix equation: THEOREM 4.20: THE INVERSE OF A TRANSITION MATRIX If P is the transition matrix from a basis B to a basis B in n R, then is invertible and the transition matrix from to is given by. CREATED BY SHANNON MARTIN GRACEY 134

135 LEMMA Let B v v v and If,,..., n 1 2 v c u c u c u n1 v c u c u c u n2 v c u c u c u n 1n 1 2n 2 nn n then the transition matrix from to is n n B u1, u2,..., u n be two bases for a vector space V. Q c c cn 1 c 11 1n nn THEOREM 4.21: TRANSITION MATRIX FROM B TO B Let B v v v and,,..., n 1 2 n B u1, u2,..., un be two bases for R. Then the transition matrix from to can be found using Gauss- Jordan elimination on the n 2n matrix B B as follows. Example 4: Find the transition matrix from B to B. 1,1, 1, 0, B 1, 0, 0,1 B CREATED BY SHANNON MARTIN GRACEY 135

136 Example 5: Find the coordinate matrix of p relative to the standard basis for 3 P. p x x CREATED BY SHANNON MARTIN GRACEY 136

137 Section 5.1: LENGTH AND DOT PRODUCT IN n R When you are done with your homework you should be able to Find the length of a vector and find a unit vector Find the distance between two vectors Find a dot product and the angle between two vectors, determine orthogonality, and verify the Cauchy-Schwartz Inequality, the triangle inequality, and the Pythagorean Theorem Use a matrix product to represent a dot product DEFINITION OF LENGTH OF A VECTOR IN n R The, or of a vector v v1, v2,..., vn in is given by When would the length of a vector equal to 0? CREATED BY SHANNON MARTIN GRACEY 137

138 Example 1: Consider the following vectors: 1 u 1, 2 a. Find u v 1 2, 2 b. Find v c. Find u v d. Find 3u e. Any observations? CREATED BY SHANNON MARTIN GRACEY 138

139 THEOREM 5.1: LENGTH OF A SCALAR MULTIPLE Let v be a vector in n R and let c be a scalar. Then where is the of c. Proof: THEOREM 5.2: UNIT VECTOR IN THE DIRECTION OF v If v is a nonzero vector in n R, then the vector has length and has the same as v. Proof: CREATED BY SHANNON MARTIN GRACEY 139

140 Example 2: Find the vector v with v 3 and the same direction as u 0, 2,1, 1. DEFINITION OF DISTANCE BETWEEN TWO VECTORS The distance between two vectors u and v in n R is CREATED BY SHANNON MARTIN GRACEY 140

141 Example 3: Find the distance between u 1,1, 2 and 1,3,0 v. CREATED BY SHANNON MARTIN GRACEY 141

142 DEFINITION OF DOT PRODUCT IN The dot product of u u u u and v v v quantity,,..., n 1 2 n R v is the,,..., n 1 2 Example 4: Consider the following vectors: u 1,2 v 2, 2 a. Find u v c. Find u 2 b. Find v v d. Find u v v e. Find u 5v CREATED BY SHANNON MARTIN GRACEY 142

143 THEOREM 5.3: PROPERTIES OF THE DOT PRODUCT If u, v and w are vectors in are true. 1. u v n R, and c is a scalar, then the following properties 2. u v w 3. cu v 4. v v 5. v v 0, and v v 0 iff. Example 5: Find 3 3 u v u v given that u u 8, u v 7, and v v 6. CREATED BY SHANNON MARTIN GRACEY 143

144 THEOREM 5.4: THE CAUCHY-SCWARZ INEQUALITY If u and v are vectors in n R, then where denotes the value of u v. Proof: CREATED BY SHANNON MARTIN GRACEY 144

145 Example 6: Verify the Cauch-Schwarz Inequality for u 1,0 and 1,1 v. DEFINITION OF THE ANGLE BETWEEN TWO VECTORS IN n R The between two nonzero vectors in n R is given by Example 7: Find the angle between u 2, 1 and 2,0 v. CREATED BY SHANNON MARTIN GRACEY 145

146 DEFINITION OF ORTHOGONAL VECTORS Two vectors u and v in n R are orthogonal if Example 8: Determine all vectors in 2 R that are orthogonal to 3,1 u. THEOREM 5.5: THE TRIANGLE INEQUALITY If u and v are vectors in n R, then Proof: CREATED BY SHANNON MARTIN GRACEY 146

147 THEOREM 5.6: THE PYTHAGOREAN THEOREM If u and v are vectors in n R, then u and v are orthogonal if and only if Example 8: Verify the Pythagoren Theorem for the vectors u 3, 2 v 4,6. and CREATED BY SHANNON MARTIN GRACEY 147

148 Section 5.2: INNER PRODUCT SPACES When you are done with your homework you should be able to Determine whether a function defines an inner product, and find the inner n product of two vectors in R, M m, n P, and C a, b Find an orthogonal projection of a vector onto another vector in an inner product space, n DEFINITION OF AN INNER PRODUCT Let u, v, and w be vectors in a vector space V, and let c be any scalar. An inner product on V is a function that associates a real number u, v with each pair of vectors u and v and satisfies the following axioms. 1. u, v 2. u, v w 3. c u, v 4. v, v 0, and v, v 0 iff NOTE: The product is an example of an product. is the inner product for. is the inner product for a space. A vector space with an inner product is called an. CREATED BY SHANNON MARTIN GRACEY 148

149 Example 1: Show that the function u, v u1v1 2u2v2 u3v3 defines an inner product on 3 R, where, u1, u2, u3 u and v, v, v v CREATED BY SHANNON MARTIN GRACEY 149

150 Example 2: Show that the function u, v u1v1 u2v2 u3v3 does not define an inner product on 3 R, where, u1, u2, u3 u and v, v, v v THEOREM 5.7: PROPERTIES OF INNER PRODUCTS Let u, v, and w be vectors in an inner product space V, and let c be any real number. 1. 0, v 2. u v, w Proof: 3. u, cv CREATED BY SHANNON MARTIN GRACEY 150

151 DEFINITION OF LENGTH, DISTANCE, AND ANGLE Let u and v be vectors in an inner product space V. 1. The length (or ) of u is. 2. The distance between u and v is. 3. The angle between two vectors u and v is given by. 4. u and v are orthogonal when. If, then u is called a vector. Moreover, if v is any nonzero vector in an inner product space V, then the vector is a vector and is called the vector in the of v. CREATED BY SHANNON MARTIN GRACEY 151

152 Inner product on C a, b is f, g. Inner product on M 2,2 is A, B. Inner product on Pn is p q, where and. Example 3: Consider the following inner product defined on u 0, 6, 1,1 v, and u, v u1v1 2u2v2 n R : a. Find u, v b. Find u c. Find v d. Find d u, v CREATED BY SHANNON MARTIN GRACEY 152

153 Example 4: Consider the following inner product defined: 1 2, f x x, f, g f x g x dx 1 a. Find f, g g x x x 2 b. Find f CREATED BY SHANNON MARTIN GRACEY 153

154 c. Find g d. Find d f, g CREATED BY SHANNON MARTIN GRACEY 154

155 THEOREM 5.8 Let u and v be vectors in an inner product space V. 1. Cauchy-Schwarz Inequality: 2. Triangle Inequality: 3. Pythagorean Theorem: u and v are orthogonal if and only if Example 5: Verify the triangle inequality for A, B a b a b a b a b A , B 2 2, and CREATED BY SHANNON MARTIN GRACEY 155

156 DEFINITION OF ORTHOGONAL PROJECTION Let u and v be vectors in an inner product space V, such that orthogonal projection of u onto v is v 0. Then the THEOREM 5.9: ORTHOGONAL PROJECTION AND DISTANCE Let u and v be vectors in an inner product space V, such that v 0. Then CREATED BY SHANNON MARTIN GRACEY 156

157 Example 6: Consider the vectors u 1, 2 and v 4, 2 following: a. proj v u. Use the Euclidean inner product to find the b. proj u v c. Sketch the graph of both proj v u and proj u v. CREATED BY SHANNON MARTIN GRACEY 157

158 Section 5.3: ORTHONORMAL BASES: GRAM-SCHMIDT PROCESS When you are done with your homework you should be able to Show that a set of vectors is orthogonal and forms an orthonormal basis, and represent a vector relative to an orthonormal basis Apply the Gram-Schmidt orthonormalization process Consider the standard basis for 3 R, which is This set is the standard basis because it has useful characteristics such as The three vectors in the basis are, and they are each. DEFINITIONS OF ORTHOGONAL AND ORTHONORMAL SETS A set S of a vector space V is called orthogonal when every pair of vectors in S is orthogonal. If, in addition, each vector in the set is a unit vector, then S is called. For the following form. S v1, v 2,..., v n, this definition has ORTHOGONAL ORTHONORMAL If is a, then it is an basis or an basis, respectively. CREATED BY SHANNON MARTIN GRACEY 158

159 THEOREM 5.10: ORTHOGONAL SETS ARE LINEARLY INDEPENDENT S v1, v 2,..., v n is an orthogonal set of vectors in an If inner product space V, then S is linearly independent. Proof: THEOREM 5.10: COROLLARY If V is an inner product space of dimension n, then any orthogonal set of n nonzero vectors is a basis for V. CREATED BY SHANNON MARTIN GRACEY 159

160 Example 1: Consider the following set in 4 R , 0, 0,, 0, 0,1, 0, 0,1, 0, 0,, 0, 0, a. Determine whether the set of vectors is orthogonal. b. If the set is orthogonal, then determine whether it is also orthonormal. c. Determine whether the set is a basis for n R. CREATED BY SHANNON MARTIN GRACEY 160

161 THEOREM 5.11: COORDINATES RELATIVE TO AN ORTHONORMAL BASIS B v1, v 2,..., v n is an orthonormal basis for an inner product space V, then If the coordinate representation of a vector w relative to B is Proof: The coordinates of relative to the basis are called the coefficients of relative to. The corresponding coordinate matrix of relative to is CREATED BY SHANNON MARTIN GRACEY 161

162 Example 2: Show that the set of vectors 2, 5, 10, 4 normalize the set to produce an orthonormal set. in 2 R is orthogonal and Example 3: Find the coordinate matrix of x 3, 4 relative to the orthonormal basis B,,, in 2 R. CREATED BY SHANNON MARTIN GRACEY 162

163 THEOREM 5.12: GRAM-SCHMIDT ORTHONORMALIZATION PROCESS 1. Let 1, 2,..., n 2. Let w B v v v be a basis for an inner product V. B w1, w 2,..., w n, where i v 1 1 v, w w v w w1, w 1 v, w v, w w v w w w1, w 1 w 2, w 2 w is given by v, w v, w v, w w v w w w 3. Let n 1 n 2 n n 1 n n 1 2 n1 w1, w 1 w 2, w 2 w n1, w n1 u i w w i i B u1, u 2,..., u n is an orthonormal basis for. Then the set v v v u u u for k 1, 2,..., n. V. Moreover, span,,..., span,,..., 1 2 k 1 2 k CREATED BY SHANNON MARTIN GRACEY 163

164 Example 4: Apply the Gram-Schmidt orthonormalization process to transform the 3 B 1, 0, 0, 1,1,1, 1,1, 1 for a subspace in R into an orthonormal basis. basis Use the Euclidean inner product on given. 3 R and use the vectors in the order they are CREATED BY SHANNON MARTIN GRACEY 164

165 Section 5.4: MATHEMATICAL MODELS AND LEAST SQUARES ANALYSIS When you are done with your homework you should be able to Define the least squares problem Find the orthogonal complement of a subspace and the projection of a vector onto a subspace Find the four fundamental subspaces of a matrix Solve a least squares problem Use least squares for mathematical modeling In this section we will study systems of linear equations and learn how to find the of such a system. LEAST SQUARES PROBLEM Given an m n matrix A and a vector b in m R, the problem is to find in m R such that is. CREATED BY SHANNON MARTIN GRACEY 165

166 FUNDAMENTAL SUBSPACES OF A MATRIX Recall that if A is an m n matrix, then the column space of A is a of consisting of all vectors of the form,. The four fundamental subspaces of the matrix A are defined as follows. = nullspace of A = nullspace of T A = column space of A = column space of T A Example 1: Find bases for the four fundamental subspaces of the matrix A CREATED BY SHANNON MARTIN GRACEY 166

167 SOLVING THE LEAST SQUARES PROBLEM Recall that we are attempting to find a vector x that minimizes, where A is an m n matrix and b is a vector in m R. Let S be the column space of A :. Assume that b is not in S, because otherwise the system Ax b would be. We are looking for a vector in that is as close as possible to. This desired vector is the of onto. So, and = is orthogonal to. However, this implies that is in, which equals. So, is in the of. The solution of the least squares problem comes down to solving the linear system of equations. These equations are called the equations of the least squares problem. CREATED BY SHANNON MARTIN GRACEY 167

168 Example 2: Find the least squares solution of the system Ax b A , 2 1 b 0 2 CREATED BY SHANNON MARTIN GRACEY 168

169 Example 3: The table shows the numbers of doctoral degrees y (in thousands) awarded in the United States from 2005 through Find the least squares regression line for the data. Then use the model to predict the number of degrees awarded in Let t represent the year, with t = 5 corresponding to (Source: U.S. National Center for Education Statistics) Year Doctoral Degrees, y CREATED BY SHANNON MARTIN GRACEY 169

170 Section 6.1: INTRODUCTION TO LINEAR TRANSFORMATIONS When you are done with your homework you should be able to Find the image and preimage of a function Show that a function is a linear transformation, and find a linear transformation IMAGES AND PREIMAGES OF FUNCTIONS In this section we will learn about functions that a vector space onto a vector space. This is denoted by. The standard function terminology is used for such functions. is called the of, and is called the of. If is v in V, and w in W such that, is called the of under. The set of all images of vectors in V is called the of, and the set of all v in V such that is called the of. Example 1: Use the function to find (a) the image of v and (b) the preimage of w., 2,,, v 0, 6, w 3,1, 2 T v v v v v v CREATED BY SHANNON MARTIN GRACEY 170

171 DEFINITION OF A LINEAR TRANSFORMATION Let V and W be vector spaces. The function T : V W is called a linear transformation of into when the following two properties are true for all u and v in V and any scalar c A linear transformation is because the same result occurs whether you perform the operations of addition and scalar multiplication or applying the. Although the same symbols denote the vector operations in both V and W, you should note that the operations may be different. Example 2: Determine whether the function is a linear transformation. a. T : R 3 R 3, T x, y, z x 1, y 1, z 1 CREATED BY SHANNON MARTIN GRACEY 171

172 b. :, T M R T A a b c d 2,2 THEOREM 6.1: PROPERTIES OF LINEAR TRANSFORMATIONS Let T be a linear transformation from V into W, where u and v are in V. Then the following properties are true Proof: 4. If, then = CREATED BY SHANNON MARTIN GRACEY 172

173 3 3 Example 3: Let T : R R be a linear transformation such that 1, 0, 0 2, 4, 1 0,1, 0 1,3, 2 T 0,0,1 0, 2, 2. Find the T, T, and indicated image. T 2, 1, 0 THEOREM 6.2: THE LINEAR TRANSFORMATION GIVEN BY A MATRIX Let A be an m n matrix. The function T defined by n m is a linear transformation from R into R. In order to conform to matrix multiplication with an m n matrix, n 1 matrices represent the vectors in m and m 1 matrices represent the vectors in R. Av a11 a1n v1 a11v1 a1nvn a a v a v a v m1 mn n m1 1 mn n n R CREATED BY SHANNON MARTIN GRACEY 173

174 n Example 4: Define the linear transformation T : R R dimensions of n R and m R. m by T v Av. Find the a. A b. A Example 5: Consider the linear transformation from Example 4, part a. a. Find T 2, 4 CREATED BY SHANNON MARTIN GRACEY 174

175 b. Find the preimage of 1, 2, 2 c. Explain why the vector 1,1,1 has no preimage under this transformation. CREATED BY SHANNON MARTIN GRACEY 175

176 Example 6: Let T be the linear transformation from P 2 into R given by the 1 integral T p p x dx. Find the preimage of 1. That is, find the polynomial 0 T p. function(s) of degree 2 or less such that 1 CREATED BY SHANNON MARTIN GRACEY 176

177 Section 6.2: THE KERNEL AND RANGE OF A LINEAR TRANSFORMATIONS When you are done with your homework you should be able to Find the kernel of a linear transformation Find a basis for the range, the rank, and the nullity of a linear transformation Determine whether a linear transformation is one-to-one or onto Determine whether vector spaces are isomorphic THE KERNEL OF A LINEAR TRANSFORMATION We know from Theorem 6.1 that for any linear transformation, the zero vector in maps to the vector in. That is,. In this section, we will consider whether there are other vectors such that. The collection of all such is called the of. Note that the zero vector is denoted by the symbol in both and, even though these two zero vectors are often different. DEFINITION OF KERNEL OF A LINEAR TRANSFORMATION Let T : V W be a linear transformation. Then the set of all vectors v in V that satisfy is called the of T and is denoted by. CREATED BY SHANNON MARTIN GRACEY 177

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