# A matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and

Save this PDF as:
Size: px
Start display at page:

Download "A matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and"

## Transcription

1 Section 5.5. Matrices and Vectors A matrix is a rectangular array of objects arranged in rows and columns. The objects are called the entries. A matrix with m rows and n columns is called an m n matrix. m n is called the size of the matrix, and the numbers m and n are its dimensions. 1

2 A matrix in which the number of rows equals the number of columns, m = n, is called a square matrix of order n. 2

3 If A is an m n matrix, then a ij denotes the element in the i th row and j th column of A: A = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a a m1 a m2 a m3 a 1n a 2n a 3n a mn. The notation A = (a ij ) also represents this display. 3

4 Special Cases: Vectors A 1 n matrix v = (a 1 a 2... a n ) also written as v = (a 1, a 2,..., a n ) is called an row vector. 4

5 An m 1 matrix v = a 1 a 2. a m is called a column vector. The entries of a row or column vector are called the components of the vector. 5

6 Arithmetic of Matrices Let A = (a ij ) be an m n matrix and let B = (b ij ) be a p q matrix. Equality: A = B if and only if 1. m = p and n = q; 2. a ij = b ij for all i and j. That is, A = B if and only if A and B are identical. 6

7 Example: a b 3 2 c 0 = 7 1 x if and only if a = 7, b = 1, c = 4, x = 3. 7

8 Addition: Let A = (a ij ) and B = (b ij ) be m n matrices. A+B is the m n matrix C = (c ij ) where c ij = a ij + b ij for all i and j. That is, A + B = (a ij + b ij ). Addition of matrices is not defined for matrices of different sizes. 8

9 Examples: (a) = (b) is not defined. 9

10 Properties: Let A, B, and C be matrices of the same size. Then: 1. A+B = B +A (Commutative) 2. (A + B) + C = A + (B + C) (Associative) A matrix with all entries equal to 0 is called a zero matrix. E.g., and

11 The symbol 0 will be used to denote the zero matrix of arbitrary size. 3. A + 0 = 0 + A = A. The zero matrix is the additive identity. 11

12 The negative of a matrix A, denoted by A is the matrix whose entries are the negatives of the entries of A. A is also called the additive inverse of A. Example: A = A =

13 4. A + ( A) = 0. Subtraction: Let A = (a ij ) and B = (b ij ) be m n matrices. Then A B = A + ( B). Example: = = 13

14 Properties: Let A, B, and C be matrices of the same size. Then: 1. A+B = B+A (commutative) 2. (A + B) + C = A + (B + C) (associative) 3. A + 0 = 0 + A = A (additive identity) 4. A + ( A) = ( A) + A = 0 (additive inverse) 14

15 Multiplication of a Matrix by a Number The product of a number k and a matrix A, denoted ka, is given by ka = (ka ij ). This product is also called multiplication by a scalar. 15

16 Example: = 16

17 Properties: Let A, B be m n matrices and let α, β be real numbers. Then 1. 1 A = A 2. 0 A = 0 3. α(a + B) = α A + α B 4. (α + β)a = α A + β A 17

18 Matrix Multiplication 1. The Product of a Row Vector and a Column Vector: The product of a 1 n row vector and an n 1 column vector is the number given by (a 1, a 2, a 3,..., a n ) b 1 b 2 b 3. b n = a 1 b 1 + a 2 b 2 + a 3 b a n b n. 18

19 Also called scalar product (because the result is a number (scalar)), dot product, and inner product. The product of a row vector and a column vector (of the same dimension and in that order!) is a number. The product of a row vector and a column vector of different dimensions is not defined. 19

20 Examples: (3, 2, 5) ( 2, 3, 1, 4)

21 2. The Product of Two Matrices: Let A = (a ij ) be an m p matrix and let B = (b ij ) be a p n matrix. The matrix product of A and B (in that order), denoted AB, is the m n matrix C = (c ij ), where c ij is the product of the i th row of A and the j th column of B. 21

22 Let A and B be given matrices. The product AB, in that order, is defined if and only if the number of columns of A equals the number of rows of B. If the product AB is defined, then the size of the product is: (no. of rows of A) (no. of columns of B): A m p B = C p n m n 22

23 Examples: 1. A = , B = AB = = BA = = 23

24 2. A = , B = AB = BA does not exist. 24

25 3. A = , B = AB = BA = Conclusion: Matrix multiplication is not commutative; AB BA in general. 25

26 Properties of Matrix Multiplication: Let A, B, and C be matrices. 1. AB BA in general; NOT COMMUTATIVE. 2. (AB)C = A(BC) matrix multiplication is associative. 26

27 Identity Matrices: Let A be a square matrix of order n. The entries a 11, a 22, a 33,..., a nn form the main diagonal of A. For each positive integer n > 1, let I n denote the square matrix of order n whose entries on the main diagonal are all 1, and all other entries are 0. The matrix I n is called the n n identity matrix. 27

28 I 2 = , I 3 = , I 4 = and so on. 28

29 3. If A is an m n matrix, then I m A = A and AI n = A. If A is a square matrix of order n, then AI n = I n A = A, 4. Inverse????? 29

30 Distributive Laws: 1. A(B + C) = AB + AC. This is called the left distributive law. 2. (A+B)C = AC + BC. This is called the right distributive law. 3. k(ab) = (ka)b = A(kB) 30

31 Other ways to look at systems of linear equations. A system of m linear equations in n unknowns: a 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b 2 a 31 x 1 + a 32 x a 3n x n = b 3... a m1 x 1 + a m2 x a mn x n = b m 31

32 Because of the way we defined multiplication, a 11 a 12 a 13 a 1n a 21 a 22 a 23 a 2n a 31 a 32 a a 3n. a m1 a m2 a m3 a mn x 1 x 2 x 3. x n = b 1 b 2 b 3. b m or in the vector-matrix form Ax = b c.f. ax = b. (1) Solution: x = A 1 b????? 32

33 Section 5.6. Square matrices 1. Inverse Let A be an n n matrix. An n n matrix B with the property that AB = BA = I n is called the multiplicative inverse of A or, more simply, the inverse of A. 33

34 Uniqueness: If A has an inverse, then it is unique. That is, there is one and only one matrix B such that AB = BA = I. B is denoted by A 1. 34

35 Procedure for finding the inverse: A = We want x y z w =

36

37 Examples: 1. A = A 1 =

38 2. B = B does not have an inverse. 37

39 NOTE: Not every nonzero n n matrix A has an inverse! 38

40 3. C = Form the augmented matrix:

41

42 C 1 =

43 Finding the inverse of A. Let A be an n n matrix. a. Form the augmented matrix (A I n ). b. Reduce (A I n ) to reduced row echelon form. If the reduced row echelon form is (I n B), then B = A 1 If the reduced row echelon form is not (I n B), then A does not have an inverse. 41

44 That is, if the reduced row echelon form of A is not the identity, then A does not have an inverse. 42

45 Application: Solve the system x +2z = 3 2x y +3z = 5 4x +y +8z = 2 Written in matrix form: x y z = Solution: Cx = b; x = C 1 b x y z = =

46 Note: The n n system of equations Ax = b has a unique solution if and only if the matrix of coefficients, A, has an inverse. 44

47 2. Determinants Find the solution set: ax + by = α cx + dy = β a b c d x y = α β, Ax = b (ad bc)x = (αd βb) (ad bc)y = (aβ cα) The number ad bc is called the determinant of A, denoted det A. 45

48 A. Calculations A = (a) det A = a A = det A = a b c d a b c d = ad bc. 46

49 A = a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 deta = a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 = a 1 b 2 c 3 a 1 b 3 c 2 a 2 b 1 c 3 +a 2 b 3 c 1 +a 3 b 1 c 2 a 3 b 2 c 1 a 1 (b 2 c 3 b 3 c 2 ) a 2 (b 1 c 3 b 3 c 1 )+a 3 (b 1 c 2 b 2 c 1 ) = a 1 b 2 b 3 c 2 c 3 a 2 b 1 b 3 c 1 c 3 + a 3 b 1 b 2 c 1 c 2 This is called the expansion across the first row. 47

50 Example: A = = ( 3) = 2(12 2) ( 3)(0+4)+3(0+8) = 2(10) ( 3)(4) + 3(8) = 56 48

51 a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 = a 1 (b 2 c 3 b 3 c 2 ) a 2 (b 1 c 3 b 3 c 1 )+a 3 (b 1 c 2 b 2 c 1 ) = a 2 (b 1 c 3 b 3 c 1 )+b 2 (a 1 c 3 a 3 c 1 ) c 2 (a 1 b 3 a 3 b 1 ) = a 2 b 1 b 3 c 1 c 3 + b 2 a 1 a 3 c 1 c 3 c 2 a 1 a 3 b 1 b 3 Expansion down the second column. 49

52 Example: A = = ( 3) = 3(4) + 4(12) 4 = 56 50

53 a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 = a 1 (b 2 c 3 b 3 c 2 ) a 2 (b 1 c 3 b 3 c 1 )+a 3 (b 1 c 2 b 2 c 1 ) = a 3 b 1 c 2 a 3 b 2 c 1 b 3 a 1 c 2 +b 3 a 2 c 1 +c 3 a 1 b 2 c 3 a 2 b 1 = a 3 (b 1 c 2 b 2 c 1 ) b 3 (a 1 c 2 a 2 c 1 )+c 3 (a 1 b 2 a 2 b 1 ) = a 3 b 1 b 2 c 1 c 2 b 3 a 1 a 2 c 1 c 2 + c 3 a 1 a 2 b 1 b 2 Expansion down the third column 51

54 and so on. You can expand across any row, or down any column. BUT: Associated with each position is an algebraic sign: For example, across the second row: a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 = = b 1 a 2 a 3 c 2 c 3 +b 2 a 1 a 3 c 1 c 3 b 3 a 1 a 2 c 1 c 2 52

55 determinants a 1 a 2 a 3 a 4 b 1 b 2 b 3 b 4 c 1 c 2 c 3 c 4 d 1 d 2 d 3 d 4 Sign chart

56 Recall: Find the solution set: ax + by = α cx + dy = β a b c d x y = α β, Ax = b (ad bc)x = αd bβ (ad bc)y = aβ αc (det A) x = αd bβ = α b β d (det A) y = aβ αc = a α c β 54

57 (det A) x = αd βb = α b β d (det A) y = aβ cα = a α c β If det A 0, then x = y = α b β d det A a α c β det A If det A = 0, then the system either has infinitely many solutions or no solutions. 55

58 Cramer s Rule. Given a system of n linear equations in n unknowns: (a square system) a 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b 2 a 31 x 1 + a 32 x a 3n x n = b 3... a n1 x 1 + a n2 x a nn x n = b n x i = det A i, (provided det A 0) det A where A i is the matrix A with the i th b. column replaced by the vector 56

59 The n n system of equations Ax = b has a unique solution if and only if det A 0. If det A = 0, then the system either has infinitely many solutions or no solutions. 57

60 Examples: 1. Given the system x + 2y + 2z = 3 2x y + z = 5 4x + y 2z = 2 Does Cramer s rule apply? det A = = 58

61 x + 2y + 2z = 3 2x y + z = 5 4x + y 2z = 2 Find y. y = = 59

62 2. Given the system 2x + 7y + 6z = 1 5x + y 2z = 7 3x + 8y + 4z = 1 det A = = 0 Cramer s Rule does not apply Does the system have infinitely many solutions or no solutions?? Compare with ax = b. 60

63 B. Properties of determinants: Let A be an n n matrix. 1. If A has a row or column of zeros, then det A = 0 Example: Expand down second column: = 0 61

64 2. If A is a diagonal matrix, A = det A = a 1 b c 3 a b c 3 0, c b = a 1 b 2 c 3. In particular, det I n = 1 For example, I 3 : = 1 62

65 3. If A is a triangular matrix, e.g., A = a 1 a 2 a 3 a 4 0 b 2 b 3 b c 3 c d 4 (upper triangular) Expand down first column!! det A = a 1 b 2 c 3 d 4 63

66 4. If B is obtained from by interchanging any two rows (columns), then det B = det A. 64

67 NOTE: If A has two identical rows (or columns), then det A = 0. 65

68 5. Multiply a row (column) of A by a nonzero number k to obtain a matrix B. Then det B = kdet A. 66

69 6. Multiply a row (column) of A by a number k and add it to another row (column) to obtain a matrix B. Then det B = det A. 67

70 7. Let A and B be n n matrices. Then det[ab] = det Adet B. 68

71 Example: Calculate

72 = =

73 Equivalences: 1. The system of equations: Ax = b has a unique solution. 2. The reduced row echelon form of A is I n. 3. The rank of A is n. 4. A has an inverse. 5. det A 0. 71

74 A is nonsingular if det A 0; A is singular if det A = 0. 72

75 Section 5.7. Vectors; Linear Dependence/Independence in R n R 2 = {(a, b) : a, b R} the plane R 3 = {(a, b, c) : a, b, c R} 3- space R 4 = {(x 1, x 2, x 3, x 4 )} 4-space and so on 73

76 Let S = {v 1,v 2,,v k } be a set of vectors in R n. The set S is linearly dependent if there exist k numbers c 1, c 2,, c k NOT ALL ZERO such that c 1 v 1 + c 2 v c k v k = 0. (c 1 v 1 + c 2 v 2 + +c k v k is a linear combination of v 1, v 2,...) 74

77 S is linearly independent if it is not linearly dependent. That is, S is linearly independent if c 1 v 1 + c 2 v c k v k = 0 implies c 1 = c 2 = = c k = 0. 75

78 The set S is linearly dependent if there exist k numbers c 1, c 2,, c k NOT ALL ZERO such that c 1 v 1 + c 2 v c k v k = 0. Another way to say this: The set S is linearly dependent if one of the vectors can be written as a linear combination of the others. 76

79 1. Two vectors v 1, v 2. Linearly dependent iff one vector is a multiple of the other. Examples: v 1 = (1, 2, 4), v 2 = ( 1, 1, 2) 2 linearly dependent: v 1 = 2v 2 77

80 v 1 = (2, 4, 5), v 2 = (0, 0, 0) linearly dependent: v 2 = 0v 1 v 1 = (5, 2, 0), v 2 = ( 3, 1, 9) linearly independent 78

81 2. v 1 = (1, 1), v 2 = (2, 3) v 3 = (3, 5) {v 1, v 2 } Dependent or independent?? {v 1, v 2, v 3 } Dependent or independent?? 79

82

83 Does there exist three numbers, c 1, c 2, c 3, not all zero such that c 1 v 1 + c 2 v 2 + c 3 v 3 = (0,0) That is, does the system of equations c 1 + 2c 2 + 3c 3 = 0 c 1 3c 2 5c 3 = 0 have nontrivial solutions? 80

84 A homogeneous system with more unknowns than equations always has infinitely many nontrivial solutions. Let v 1,v 2,,v k be a set of k vectors in R n. If k > n, then the set of vectors is (automatically) linearly dependent. 81

85 3. v 1 = (1, 1, 2), v 2 = (2, 3, 0), v 3 = ( 1, 2, 2), v 4 = (0, 4, 3) Dependent or independent?? a. {v 1, v 2, v 3, v 4 } b. {v 1, v 2, v 3 } c. {v 1, v 2 } 82

86 v 1 = (1, 1, 2), v 2 = (2, 3, 0), v 3 = ( 1, 2, 2) Does c 1 v 1 + c 2 v 2 + c 3 v 3 = 0 have non-trivial solutions? That is, does c 1 + 2c 2 c 3 = 0 c 1 3c 2 2c 3 = 0 2c 1 + 2c 3 = 0 have non-trivial solutions 83

87 Augmented matrix and row reduce:

88 NOTE: It s enough to row reduce:

89 Calculate the determinant

90 Or, row reduce

91 4. v 1 = ( a, 1, 1), v 2 = ( 1, 2a, 3), v 3 = ( 2, a, 2), v 4 = (3a, 2, a) For what values of a are the vectors linearly dependent? 88

92 4. v 1 = ( a, 1, 1), v 2 = ( 1, 2a, 3), v 3 = ( 2, a, 2), v 4 = (3a, 2, a) For what values of a are the vectors v 1, v 2, v 3 linearly dependent? a c 1 c 2 2c 3 = 0 c 1 + 2a c 2 + a c 3 = 0 c 1 + 3c 2 + 2c 3 = 0 89

93 Augmented matrix and row reduce: a a a

94 Or row reduce: a a a

95 Calculate the determinant a a 3 2 a 2 92

96 5. v 1 = (1, 1,2,1), v 2 = (3,2,0, 1) v 3 = ( 1, 4,4,3), v 4 = (2,3, 2, 2) a. {v 1, v 2, v 3, v 4 } dependent or independent? b. What is the maximum number of independent vectors? 93

97 Row reduce

98 Or, calculate the determinant

99 Tests for independence/dependence Let S = v 1,v 2,,v k be a set of vectors in R n. Case 1: k > n : S is linearly dependent. 96

100 Case 2: k = n : Form the n n matrix A whose rows are v 1, v 2,,v n 1. Row reduce A: if the reduced matrix has n nonzero rows independent; one or more zero rows dependent. 97

101 Equivalently, solve the system of equations c 1 v 1 + c 2 v c k v k = 0. Unique solution: c 1 = c 2 = = c n = 0 independent; infinitely many solutions dependent.

102 2. Calculate det A: det A 0 independent; det A = 0 dependent. Note: If v 1,v 2,,v n is a linearly independent set of vectors in R n, then each vector in R n has a unique representation as a linear combination of v 1,v 2,,v n.

103 Case 3: k < n : Form the k n matrix A whose rows are v 1, v 2,,v k 1. Row reduce A: if the reduced matrix has k nonzero rows independent; one or more zero rows dependent. 98

104 Equivalently, solve the system of equations c 1 v 1 + c 2 v c k v k = 0. Unique solution: c 1 = c 2 = = c n = 0 independent; infinitely many solutions dependent.

105 Linear Dependence/Independence of Functions Let S = {f 1 (x), f 2 (x), f k (x)} be a set of functions defined on an interval I. S is linearly dependent if there exist k numbers c 1, c 2,, c k NOT ALL ZERO such that c 1 f 1 (x)+c 2 f 2 (x)+ +c k f k (x) 0 on I. 99

106 S is linearly independent if it is not linearly dependent. That is, S is linearly independent if c 1 f 1 (x)+c 2 f 2 (x)+ +c k f k (x) 0 on I implies c 1 = c 2 = = c k = 0. Examples: 100

107 1. f 1 (x) = 1, f 2 (x) = x, f 3 (x) = x 2 Suppose f 1, f 2, f 3 are linearly dependent. Then there exist 3 numbers c 1, c 2, c 3, NOT ALL 0 such that c c 2 x + c 3 x 2 0 That is, c c 2 x + c 3 x 2 = 0 for all x 101

108 2. f 1 (x) = sin x, f 2 (x) = cos x Suppose f 1, f 2 are linearly dependent. Then there exist 2 numbers c 1, c 2, NOT BOTH 0 such that c 1 cos x + c 2 sin x 0 That is, c 1 cos x + c 2 sin x = 0 for all x 102

109 3. f 1 (x) = sin x, f 2 (x) = cos x, f 3 (x) = cos(x + π/3) 103

110 Test for independence/dependence Suppose that the functions f 1 (x), f 2 (x), f k (x) are (k 1)-times differentiable on I. If the determinant W(x) = f 1 (x) f 2 (x) f k (x) f 1 (x) f 2 (x) f k (x) f (k 1) 1 (x) f (k 1) 2 (x) f (k 1) (x) k 0 for at least one x I, then the functions are linearly independent on I. NOTE: If W(x) 0 in I, then?????? The functions may be dependent, they may be independent!!!! 104

111 W(x) is called THEEEEEE

112 WRONSKIAN 105

113 Examples: f 1 = 2x+1, f 2 = x 3, f 3 = 3x+2 Wronskian: 2x + 1 x 3 3x = 0 Independent or dependent??? Does there exist 3 numbers, c 1, c 2, c 3, not all 0, such that c 1 f 1 + c 2 f 2 + c 3 f

114

115 f 1 (x) = x 3, f 2 (x) = x 3, x < 0 2x 3, x 0 Wronskian: For x < 0 x 3 x 3 3x 2 3x 2 = 0 For x 0 x 3 2x 3 3x 2 6x 2 = 0 107

116 Suppose there exist 2 numbers c 1, c 2, not both zero, such that c 1 f 1 (x) + c 2 f 2 (x) 0 108

117 Interpretations of a system of linear equations Given the system of m linear equations in n unknowns: a 11 x 1 + a 12 x 2 + a 13 x a 1n x n = b 1 a 21 x 1 + a 22 x 2 + a 23 x a 2n x n = b 2 a 31 x 1 + a 32 x 2 + a 33 x a 3n x n = b 3... a m1 x 1 + a m2 x 2 + a m3 x a mn x n = b m 109

118 The coefficient matrix A from R n to R m is a mapping a 11 a 12 a 13 a 1n a 21 a 22 a 23 a 2n a 31. a 32. a 33.. a 3n a m1 a m2 a m3 a mn x 1 x 2 x 3. x n = b 1 b 2 b 3. b m 110

119 A is a linear transformation! A[x + y] = Ax + Ay and A[αx] = α Ax Example: A = ( ) is a linear transformation from R 3 to R

120 Write the system as a 11 a 12 a 13 a 1n a 21 a 22 a 23 a 2n a 31. a 32. a 33.. a 3n a m1 a m2 a m3 a mn x 1 x 2 x 3. x n = b 1 b 2 b 3. b m or in the vector-matrix form Ax = b Given a vector b in R m, find a vector x in R n such that Ax = b. 112

121 Write the system equivalently as x 1 a 11 a 21. +x 2 a 12 a x n a 1n a 2ṇ. = b 1 b 2. a m1 a m2 a mn b m This says, express b as a linear combination of the columns of A. 113

122 Section 5.8. Eigenvalues/Eigenvectors Example: Set A = A is a linear transformation from R 3 to R =

123 = = = = =

124 Let A be an n n matrix. A number λ is an eigenvalue of A if there is a non-zero vector v such that Av = λv 116

125 To find the eigenvalues of A, find the values of λ that satisfy det(a λ I) = 0. Example: Let A = A λi = 2 λ λ det (A λi) = 2 λ λ = (2 λ)(4 λ) 3 = λ 2 6λ+5 = 0 Eigenvalues: λ 1 = 5, λ 2 = 1 117

126 Let A = det (A λi) = 1 λ λ λ = λ 3 + λ 2 + 4λ 4 det (A λi) = 0 implies λ 3 + λ 2 + 4λ 4 = 0 or λ 3 λ 2 4λ + 4 = 0 118

127 Terminology: det(a λ I) is a polynomial of degree n, called the characteristic polynomial of A. The zeros of the characteristic polynomial are the eigenvalues of A The equation det(a λi) = 0 is called the characteristic equation of A. 119

128 A non-zero vector v that satisfies Av = λv is called an eigenvector corresponding to the eigenvalue λ.

129 Examples: 1. A = Characteristic polynomial: det(a λ I) = 2 λ λ = (2 λ)( 1 λ) 4 = λ 2 λ 6 Characteristic equation: λ 2 λ 6 = (λ 3)(λ + 2) = 0 Eigenvalues: λ 1 = 3, λ 2 = 2 120

130 Eigenvectors: 121

131

132 2. A = Characteristic polynomial: det(a λ I) = 4 λ 1 4 λ = λ 2 4λ + 4 Characteristic equation: λ 2 4λ + 4 = (λ 2) 2 = 0 Eigenvalues: λ 1 = λ 2 = 2 122

133 Eigenvectors: 123

134 3. A = Characteristic polynomial: det(a λ I) = 5 λ λ = λ 2 4λ + 13 Characteristic equation: λ 2 4λ + 13 = 0 Eigenvalues: λ 1 = 2 + 3i, λ 2 = 2 3i 124

135 Eigenvectors: 125

136

137 NOTE: If a + b i is an eigenvalue of A with eigenvector α+β i, then a b i is also an eigenvalue of A and α β i is a corresponding eigenvector. 126

138 4. A = Characteristic polynomial: det(a λ I) = 4 λ λ λ = λ 3 + 4λ 2 λ 6 Characteristic equation: λ 3 4λ 2 +λ+6 = (λ 3)(λ 2)(λ+1) = 0. Eigenvalues: λ 1 = 3, λ 2 = 2, λ 3 = 1 127

139 Eigenvectors: 128

140

141

142 5. A = Characteristic polynomial: det(a λ I) = 4 λ λ λ = λ λ 2 39λ + 45 Characteristic equation: λ 3 11λ 2 +39λ 45 = (λ 3) 2 (λ 5) = 0. Eigenvalues: λ 1 = λ 2 = 3, λ 3 = 5 129

143 Eigenvectors: 130

144

145 6. A = Characteristic polynomial: det(a λ I) = 3 λ λ λ = λ λ + 16 Characteristic equation: λ 3 12λ 16 = (λ+2) 2 (λ 4) = 0. Eigenvalues: λ 1 = λ 2 = 2, λ 3 = 4 131

146 Eigenvectors: 132

147

148 Some Facts THEOREM If v 1, v 2,..., v k are eigenvectors of a matrix A corresponding to distinct eigenvalues λ 1, λ 2,..., λ k, then v 1, v 2,..., v k are linearly independent. 133

149 THEOREM Let A be a (real) n n matrix. If the complex number λ = a + bi is an eigenvalue of A with corresponding (complex) eigenvector u+iv, then λ = a bi, the conjugate of a + bi, is also an eigenvalue of A and u iv is a corresponding eigenvector. 134

150 THEOREM Let A be an n n matrix with eigenvalues λ 1, λ 2,..., λ n. Then det A = ( 1) n λ 1 λ 2 λ 3 λ n. That is, det A is the product of the eigenvalues of A. (The λ s are not necessarily distinct, the multiplicity of an eigenvalue may be greater than 1, and they are not necessarily real.) 135

151 Equivalences: (A an n n matrix) 1. Ax = b has a unique solution. 2. The reduced row echelon form of A is I n. 3. The rank of A is n. 4. A has an inverse. 5. det A is not an eigenvalue of A 136

### A matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and

Section 5.5. Matrices and Vectors A matrix is a rectangular array of objects arranged in rows and columns. The objects are called the entries. A matrix with m rows and n columns is called an m n matrix.

### Chapter 5. Linear Algebra. A linear (algebraic) equation in. unknowns, x 1, x 2,..., x n, is. an equation of the form

Chapter 5. Linear Algebra A linear (algebraic) equation in n unknowns, x 1, x 2,..., x n, is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b where a 1, a 2,..., a n and b are real numbers. 1

### Matrices and Linear Algebra

Contents Quantitative methods for Economics and Business University of Ferrara Academic year 2017-2018 Contents 1 Basics 2 3 4 5 Contents 1 Basics 2 3 4 5 Contents 1 Basics 2 3 4 5 Contents 1 Basics 2

### Linear Algebra. Matrices Operations. Consider, for example, a system of equations such as x + 2y z + 4w = 0, 3x 4y + 2z 6w = 0, x 3y 2z + w = 0.

Matrices Operations Linear Algebra Consider, for example, a system of equations such as x + 2y z + 4w = 0, 3x 4y + 2z 6w = 0, x 3y 2z + w = 0 The rectangular array 1 2 1 4 3 4 2 6 1 3 2 1 in which the

### [ Here 21 is the dot product of (3, 1, 2, 5) with (2, 3, 1, 2), and 31 is the dot product of

. Matrices A matrix is any rectangular array of numbers. For example 3 5 6 4 8 3 3 is 3 4 matrix, i.e. a rectangular array of numbers with three rows four columns. We usually use capital letters for matrices,

### Chapter 7. Linear Algebra: Matrices, Vectors,

Chapter 7. Linear Algebra: Matrices, Vectors, Determinants. Linear Systems Linear algebra includes the theory and application of linear systems of equations, linear transformations, and eigenvalue problems.

### Equality: Two matrices A and B are equal, i.e., A = B if A and B have the same order and the entries of A and B are the same.

Introduction Matrix Operations Matrix: An m n matrix A is an m-by-n array of scalars from a field (for example real numbers) of the form a a a n a a a n A a m a m a mn The order (or size) of A is m n (read

### 3 Matrix Algebra. 3.1 Operations on matrices

3 Matrix Algebra A matrix is a rectangular array of numbers; it is of size m n if it has m rows and n columns. A 1 n matrix is a row vector; an m 1 matrix is a column vector. For example: 1 5 3 5 3 5 8

### MATH 3321 Sample Questions for Exam 3. 3y y, C = Perform the indicated operations, if possible: (a) AC (b) AB (c) B + AC (d) CBA

MATH 33 Sample Questions for Exam 3. Find x and y so that x 4 3 5x 3y + y = 5 5. x = 3/7, y = 49/7. Let A = 3 4, B = 3 5, C = 3 Perform the indicated operations, if possible: a AC b AB c B + AC d CBA AB

### Lecture Notes in Linear Algebra

Lecture Notes in Linear Algebra Dr. Abdullah Al-Azemi Mathematics Department Kuwait University February 4, 2017 Contents 1 Linear Equations and Matrices 1 1.2 Matrices............................................

### Elementary Row Operations on Matrices

King Saud University September 17, 018 Table of contents 1 Definition A real matrix is a rectangular array whose entries are real numbers. These numbers are organized on rows and columns. An m n matrix

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 1 x 2. x n 8 (4) 3 4 2

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

### Study Guide for Linear Algebra Exam 2

Study Guide for Linear Algebra Exam 2 Term Vector Space Definition A Vector Space is a nonempty set V of objects, on which are defined two operations, called addition and multiplication by scalars (real

### Chapter 1: Systems of linear equations and matrices. Section 1.1: Introduction to systems of linear equations

Chapter 1: Systems of linear equations and matrices Section 1.1: Introduction to systems of linear equations Definition: A linear equation in n variables can be expressed in the form a 1 x 1 + a 2 x 2

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

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

### Online Exercises for Linear Algebra XM511

This document lists the online exercises for XM511. The section ( ) numbers refer to the textbook. TYPE I are True/False. Lecture 02 ( 1.1) Online Exercises for Linear Algebra XM511 1) The matrix [3 2

### POLI270 - Linear Algebra

POLI7 - Linear Algebra Septemer 8th Basics a x + a x +... + a n x n b () is the linear form where a, b are parameters and x n are variables. For a given equation such as x +x you only need a variable and

### Chapter 4 - MATRIX ALGEBRA. ... a 2j... a 2n. a i1 a i2... a ij... a in

Chapter 4 - MATRIX ALGEBRA 4.1. Matrix Operations A a 11 a 12... a 1j... a 1n a 21. a 22.... a 2j... a 2n. a i1 a i2... a ij... a in... a m1 a m2... a mj... a mn The entry in the ith row and the jth column

### IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET

IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each

### MAT Linear Algebra Collection of sample exams

MAT 342 - Linear Algebra Collection of sample exams A-x. (0 pts Give the precise definition of the row echelon form. 2. ( 0 pts After performing row reductions on the augmented matrix for a certain system

### Math Linear Algebra Final Exam Review Sheet

Math 15-1 Linear Algebra Final Exam Review Sheet Vector Operations Vector addition is a component-wise operation. Two vectors v and w may be added together as long as they contain the same number n of

### 1. In this problem, if the statement is always true, circle T; otherwise, circle F.

Math 1553, Extra Practice for Midterm 3 (sections 45-65) Solutions 1 In this problem, if the statement is always true, circle T; otherwise, circle F a) T F If A is a square matrix and the homogeneous equation

### IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET

IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each

### Math 360 Linear Algebra Fall Class Notes. a a a a a a. a a a

Math 360 Linear Algebra Fall 2008 9-10-08 Class Notes Matrices As we have already seen, a matrix is a rectangular array of numbers. If a matrix A has m columns and n rows, we say that its dimensions are

### Chapter 2:Determinants. Section 2.1: Determinants by cofactor expansion

Chapter 2:Determinants Section 2.1: Determinants by cofactor expansion [ ] a b Recall: The 2 2 matrix is invertible if ad bc 0. The c d ([ ]) a b function f = ad bc is called the determinant and it associates

### A = 3 B = A 1 1 matrix is the same as a number or scalar, 3 = [3].

Appendix : A Very Brief Linear ALgebra Review Introduction Linear Algebra, also known as matrix theory, is an important element of all branches of mathematics Very often in this course we study the shapes

### and let s calculate the image of some vectors under the transformation T.

Chapter 5 Eigenvalues and Eigenvectors 5. Eigenvalues and Eigenvectors Let T : R n R n be a linear transformation. Then T can be represented by a matrix (the standard matrix), and we can write T ( v) =

### A VERY BRIEF LINEAR ALGEBRA REVIEW for MAP 5485 Introduction to Mathematical Biophysics Fall 2010

A VERY BRIEF LINEAR ALGEBRA REVIEW for MAP 5485 Introduction to Mathematical Biophysics Fall 00 Introduction Linear Algebra, also known as matrix theory, is an important element of all branches of mathematics

### MATH2210 Notebook 2 Spring 2018

MATH2210 Notebook 2 Spring 2018 prepared by Professor Jenny Baglivo c Copyright 2009 2018 by Jenny A. Baglivo. All Rights Reserved. 2 MATH2210 Notebook 2 3 2.1 Matrices and Their Operations................................

### MATRICES. a m,1 a m,n A =

MATRICES Matrices are rectangular arrays of real or complex numbers With them, we define arithmetic operations that are generalizations of those for real and complex numbers The general form a matrix of

### Math Camp II. Basic Linear Algebra. Yiqing Xu. Aug 26, 2014 MIT

Math Camp II Basic Linear Algebra Yiqing Xu MIT Aug 26, 2014 1 Solving Systems of Linear Equations 2 Vectors and Vector Spaces 3 Matrices 4 Least Squares Systems of Linear Equations Definition A linear

### Linear Systems and Matrices

Department of Mathematics The Chinese University of Hong Kong 1 System of m linear equations in n unknowns (linear system) a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.......

### 1. Let m 1 and n 1 be two natural numbers such that m > n. Which of the following is/are true?

. Let m and n be two natural numbers such that m > n. Which of the following is/are true? (i) A linear system of m equations in n variables is always consistent. (ii) A linear system of n equations in

### Review of Linear Algebra

Review of Linear Algebra Definitions An m n (read "m by n") matrix, is a rectangular array of entries, where m is the number of rows and n the number of columns. 2 Definitions (Con t) A is square if m=

### Introduction. Vectors and Matrices. Vectors [1] Vectors [2]

Introduction Vectors and Matrices Dr. TGI Fernando 1 2 Data is frequently arranged in arrays, that is, sets whose elements are indexed by one or more subscripts. Vector - one dimensional array Matrix -

### We could express the left side as a sum of vectors and obtain the Vector Form of a Linear System: a 12 a x n. a m2

Week 22 Equations, Matrices and Transformations Coefficient Matrix and Vector Forms of a Linear System Suppose we have a system of m linear equations in n unknowns a 11 x 1 + a 12 x 2 + + a 1n x n b 1

### c c c c c c c c c c a 3x3 matrix C= has a determinant determined by

Linear Algebra Determinants and Eigenvalues Introduction: Many important geometric and algebraic properties of square matrices are associated with a single real number revealed by what s known as the determinant.

### Eigenvalues and Eigenvectors

5 Eigenvalues and Eigenvectors 5.2 THE CHARACTERISTIC EQUATION DETERMINANATS nn Let A be an matrix, let U be any echelon form obtained from A by row replacements and row interchanges (without scaling),

### Introduction to Matrices

POLS 704 Introduction to Matrices Introduction to Matrices. The Cast of Characters A matrix is a rectangular array (i.e., a table) of numbers. For example, 2 3 X 4 5 6 (4 3) 7 8 9 0 0 0 Thismatrix,with4rowsand3columns,isoforder

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

MATH 20F: LINEAR ALGEBRA LECTURE B00 (T KEMP) Definition 01 If T (x) = Ax is a linear transformation from R n to R m then Nul (T ) = {x R n : T (x) = 0} = Nul (A) Ran (T ) = {Ax R m : x R n } = {b R m

### Finite Mathematics Chapter 2. where a, b, c, d, h, and k are real numbers and neither a and b nor c and d are both zero.

Finite Mathematics Chapter 2 Section 2.1 Systems of Linear Equations: An Introduction Systems of Equations Recall that a system of two linear equations in two variables may be written in the general form

### LU Factorization. A m x n matrix A admits an LU factorization if it can be written in the form of A = LU

LU Factorization A m n matri A admits an LU factorization if it can be written in the form of Where, A = LU L : is a m m lower triangular matri with s on the diagonal. The matri L is invertible and is

### Fundamentals of Engineering Analysis (650163)

Philadelphia University Faculty of Engineering Communications and Electronics Engineering Fundamentals of Engineering Analysis (6563) Part Dr. Omar R Daoud Matrices: Introduction DEFINITION A matrix is

### Fundamentals of Linear Algebra. Marcel B. Finan Arkansas Tech University c All Rights Reserved

Fundamentals of Linear Algebra Marcel B. Finan Arkansas Tech University c All Rights Reserved 2 PREFACE Linear algebra has evolved as a branch of mathematics with wide range of applications to the natural

### Linear Algebra Homework and Study Guide

Linear Algebra Homework and Study Guide Phil R. Smith, Ph.D. February 28, 20 Homework Problem Sets Organized by Learning Outcomes Test I: Systems of Linear Equations; Matrices Lesson. Give examples of

### Chapter 2 Notes, Linear Algebra 5e Lay

Contents.1 Operations with Matrices..................................1.1 Addition and Subtraction.............................1. Multiplication by a scalar............................ 3.1.3 Multiplication

### Topic 1: Matrix diagonalization

Topic : Matrix diagonalization Review of Matrices and Determinants Definition A matrix is a rectangular array of real numbers a a a m a A = a a m a n a n a nm The matrix is said to be of order n m if it

### LINEAR ALGEBRA SUMMARY SHEET.

LINEAR ALGEBRA SUMMARY SHEET RADON ROSBOROUGH https://intuitiveexplanationscom/linear-algebra-summary-sheet/ This document is a concise collection of many of the important theorems of linear algebra, organized

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

MATH 240 Spring, 2006 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 8th Ed. Sections 1.1 1.6, 2.1 2.2, 3.2 3.8, 4.3 4.5, 5.1 5.3, 5.5, 6.1 6.5, 7.1 7.2, 7.4 DEFINITIONS Chapter 1: Linear

### Ma 227 Review for Systems of DEs

Ma 7 Review for Systems of DEs Matrices Basic Properties Addition and subtraction: Let A a ij mn and B b ij mn.then A B a ij b ij mn 3 A 6 B 6 4 7 6 A B 6 4 3 7 6 6 7 3 Scaler Multiplication: Let k be

### Chapter 3. Determinants and Eigenvalues

Chapter 3. Determinants and Eigenvalues 3.1. Determinants With each square matrix we can associate a real number called the determinant of the matrix. Determinants have important applications to the theory

### ENGI 9420 Lecture Notes 2 - Matrix Algebra Page Matrix operations can render the solution of a linear system much more efficient.

ENGI 940 Lecture Notes - Matrix Algebra Page.0. Matrix Algebra A linear system of m equations in n unknowns, a x + a x + + a x b (where the a ij and i n n a x + a x + + a x b n n a x + a x + + a x b m

### Solving a system by back-substitution, checking consistency of a system (no rows of the form

MATH 520 LEARNING OBJECTIVES SPRING 2017 BROWN UNIVERSITY SAMUEL S. WATSON Week 1 (23 Jan through 27 Jan) Definition of a system of linear equations, definition of a solution of a linear system, elementary

### Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition

Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Prof. Tesler Math 283 Fall 2016 Also see the separate version of this with Matlab and R commands. Prof. Tesler Diagonalizing

### OHSx XM511 Linear Algebra: Solutions to Online True/False Exercises

This document gives the solutions to all of the online exercises for OHSx XM511. The section ( ) numbers refer to the textbook. TYPE I are True/False. Answers are in square brackets [. Lecture 02 ( 1.1)

### Matrix Algebra Determinant, Inverse matrix. Matrices. A. Fabretti. Mathematics 2 A.Y. 2015/2016. A. Fabretti Matrices

Matrices A. Fabretti Mathematics 2 A.Y. 2015/2016 Table of contents Matrix Algebra Determinant Inverse Matrix Introduction A matrix is a rectangular array of numbers. The size of a matrix is indicated

### Section 9.2: Matrices. Definition: A matrix A consists of a rectangular array of numbers, or elements, arranged in m rows and n columns.

Section 9.2: Matrices Definition: A matrix A consists of a rectangular array of numbers, or elements, arranged in m rows and n columns. That is, a 11 a 12 a 1n a 21 a 22 a 2n A =...... a m1 a m2 a mn A

### Calculating determinants for larger matrices

Day 26 Calculating determinants for larger matrices We now proceed to define det A for n n matrices A As before, we are looking for a function of A that satisfies the product formula det(ab) = det A det

### Linear Algebra Practice Problems

Math 7, Professor Ramras Linear Algebra Practice Problems () Consider the following system of linear equations in the variables x, y, and z, in which the constants a and b are real numbers. x y + z = a

### Systems of Linear Equations and Matrices

Chapter 1 Systems of Linear Equations and Matrices System of linear algebraic equations and their solution constitute one of the major topics studied in the course known as linear algebra. In the first

### MATH 213 Linear Algebra and ODEs Spring 2015 Study Sheet for Midterm Exam. Topics

MATH 213 Linear Algebra and ODEs Spring 2015 Study Sheet for Midterm Exam This study sheet will not be allowed during the test Books and notes will not be allowed during the test Calculators and cell phones

### Section 12.4 Algebra of Matrices

244 Section 2.4 Algebra of Matrices Before we can discuss Matrix Algebra, we need to have a clear idea what it means to say that two matrices are equal. Let's start a definition. Equal Matrices Two matrices

### Chapter 5. Linear Algebra. Sections A linear (algebraic) equation in. unknowns, x 1, x 2,..., x n, is. an equation of the form

Chapter 5. Linear Algebra Sections 5.1 5.3 A linear (algebraic) equation in n unknowns, x 1, x 2,..., x n, is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b where a 1, a 2,..., a n and b are

### 1. Linear systems of equations. Chapters 7-8: Linear Algebra. Solution(s) of a linear system of equations (continued)

1 A linear system of equations of the form Sections 75, 78 & 81 a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2 a m1 x 1 + a m2 x 2 + + a mn x n = b m can be written in matrix

### Eigenvalues and Eigenvectors

5 Eigenvalues and Eigenvectors 5.2 THE CHARACTERISTIC EQUATION DETERMINANATS n n Let A be an matrix, let U be any echelon form obtained from A by row replacements and row interchanges (without scaling),

### Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition

Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Prof. Tesler Math 283 Fall 2018 Also see the separate version of this with Matlab and R commands. Prof. Tesler Diagonalizing

### Vectors and matrices: matrices (Version 2) This is a very brief summary of my lecture notes.

Vectors and matrices: matrices (Version 2) This is a very brief summary of my lecture notes Matrices and linear equations A matrix is an m-by-n array of numbers A = a 11 a 12 a 13 a 1n a 21 a 22 a 23 a

### Knowledge Discovery and Data Mining 1 (VO) ( )

Knowledge Discovery and Data Mining 1 (VO) (707.003) Review of Linear Algebra Denis Helic KTI, TU Graz Oct 9, 2014 Denis Helic (KTI, TU Graz) KDDM1 Oct 9, 2014 1 / 74 Big picture: KDDM Probability Theory

### Chapter 5. Linear Algebra. A linear (algebraic) equation in. unknowns, x 1, x 2,..., x n, is. an equation of the form

Chapter 5. Linear Algebra A linear (algebraic) equation in n unknowns, x 1, x 2,..., x n, is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b where a 1, a 2,..., a n and b are real numbers. 1

### Systems of Linear Equations and Matrices

Chapter 1 Systems of Linear Equations and Matrices System of linear algebraic equations and their solution constitute one of the major topics studied in the course known as linear algebra. In the first

### Numerical Linear Algebra Homework Assignment - Week 2

Numerical Linear Algebra Homework Assignment - Week 2 Đoàn Trần Nguyên Tùng Student ID: 1411352 8th October 2016 Exercise 2.1: Show that if a matrix A is both triangular and unitary, then it is diagonal.

### Elementary Linear Algebra

Matrices J MUSCAT Elementary Linear Algebra Matrices Definition Dr J Muscat 2002 A matrix is a rectangular array of numbers, arranged in rows and columns a a 2 a 3 a n a 2 a 22 a 23 a 2n A = a m a mn We

### Linear Algebra: Lecture notes from Kolman and Hill 9th edition.

Linear Algebra: Lecture notes from Kolman and Hill 9th edition Taylan Şengül March 20, 2019 Please let me know of any mistakes in these notes Contents Week 1 1 11 Systems of Linear Equations 1 12 Matrices

### Systems of Algebraic Equations and Systems of Differential Equations

Systems of Algebraic Equations and Systems of Differential Equations Topics: 2 by 2 systems of linear equations Matrix expression; Ax = b Solving 2 by 2 homogeneous systems Functions defined on matrices

### ELEMENTARY LINEAR ALGEBRA

ELEMENTARY LINEAR ALGEBRA K. R. MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND Second Online Version, December 1998 Comments to the author at krm@maths.uq.edu.au Contents 1 LINEAR EQUATIONS

### MATH 106 LINEAR ALGEBRA LECTURE NOTES

MATH 6 LINEAR ALGEBRA LECTURE NOTES FALL - These Lecture Notes are not in a final form being still subject of improvement Contents Systems of linear equations and matrices 5 Introduction to systems of

### TOPIC III LINEAR ALGEBRA

[1] Linear Equations TOPIC III LINEAR ALGEBRA (1) Case of Two Endogenous Variables 1) Linear vs. Nonlinear Equations Linear equation: ax + by = c, where a, b and c are constants. 2 Nonlinear equation:

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

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

### Lecture Summaries for Linear Algebra M51A

These lecture summaries may also be viewed online by clicking the L icon at the top right of any lecture screen. Lecture Summaries for Linear Algebra M51A refers to the section in the textbook. Lecture

### Linear Algebra: Matrix Eigenvalue Problems

CHAPTER8 Linear Algebra: Matrix Eigenvalue Problems Chapter 8 p1 A matrix eigenvalue problem considers the vector equation (1) Ax = λx. 8.0 Linear Algebra: Matrix Eigenvalue Problems Here A is a given

### Glossary of Linear Algebra Terms. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Glossary of Linear Algebra Terms Basis (for a subspace) A linearly independent set of vectors that spans the space Basic Variable A variable in a linear system that corresponds to a pivot column in the

### Section 9.2: Matrices.. a m1 a m2 a mn

Section 9.2: Matrices Definition: A matrix is a rectangular array of numbers: a 11 a 12 a 1n a 21 a 22 a 2n A =...... a m1 a m2 a mn In general, a ij denotes the (i, j) entry of A. That is, the entry in

### MTH 464: Computational Linear Algebra

MTH 464: Computational Linear Algebra Lecture Outlines Exam 2 Material Prof. M. Beauregard Department of Mathematics & Statistics Stephen F. Austin State University March 2, 2018 Linear Algebra (MTH 464)

### Determinants by Cofactor Expansion (III)

Determinants by Cofactor Expansion (III) Comment: (Reminder) If A is an n n matrix, then the determinant of A can be computed as a cofactor expansion along the jth column det(a) = a1j C1j + a2j C2j +...

### Introduction to Matrix Algebra

Introduction to Matrix Algebra August 18, 2010 1 Vectors 1.1 Notations A p-dimensional vector is p numbers put together. Written as x 1 x =. x p. When p = 1, this represents a point in the line. When p

### Math Camp Lecture 4: Linear Algebra. Xiao Yu Wang. Aug 2010 MIT. Xiao Yu Wang (MIT) Math Camp /10 1 / 88

Math Camp 2010 Lecture 4: Linear Algebra Xiao Yu Wang MIT Aug 2010 Xiao Yu Wang (MIT) Math Camp 2010 08/10 1 / 88 Linear Algebra Game Plan Vector Spaces Linear Transformations and Matrices Determinant

### (b) If a multiple of one row of A is added to another row to produce B then det(b) =det(a).

.(5pts) Let B = 5 5. Compute det(b). (a) (b) (c) 6 (d) (e) 6.(5pts) Determine which statement is not always true for n n matrices A and B. (a) If two rows of A are interchanged to produce B, then det(b)

### Introduction to Matrices and Linear Systems Ch. 3

Introduction to Matrices and Linear Systems Ch. 3 Doreen De Leon Department of Mathematics, California State University, Fresno June, 5 Basic Matrix Concepts and Operations Section 3.4. Basic Matrix Concepts

### 1. What is the determinant of the following matrix? a 1 a 2 4a 3 2a 2 b 1 b 2 4b 3 2b c 1. = 4, then det

What is the determinant of the following matrix? 3 4 3 4 3 4 4 3 A 0 B 8 C 55 D 0 E 60 If det a a a 3 b b b 3 c c c 3 = 4, then det a a 4a 3 a b b 4b 3 b c c c 3 c = A 8 B 6 C 4 D E 3 Let A be an n n matrix

### Introduction to Matrices

214 Analysis and Design of Feedback Control Systems Introduction to Matrices Derek Rowell October 2002 Modern system dynamics is based upon a matrix representation of the dynamic equations governing the

### MTH501- Linear Algebra MCQS MIDTERM EXAMINATION ~ LIBRIANSMINE ~

MTH501- Linear Algebra MCQS MIDTERM EXAMINATION ~ LIBRIANSMINE ~ Question No: 1 (Marks: 1) If for a linear transformation the equation T(x) =0 has only the trivial solution then T is One-to-one Onto Question

### Chapter Two Elements of Linear Algebra

Chapter Two Elements of Linear Algebra Previously, in chapter one, we have considered single first order differential equations involving a single unknown function. In the next chapter we will begin to

### Math 240 Calculus III

The Calculus III Summer 2015, Session II Wednesday, July 8, 2015 Agenda 1. of the determinant 2. determinants 3. of determinants What is the determinant? Yesterday: Ax = b has a unique solution when A

### Chapter 5 Eigenvalues and Eigenvectors

Chapter 5 Eigenvalues and Eigenvectors Outline 5.1 Eigenvalues and Eigenvectors 5.2 Diagonalization 5.3 Complex Vector Spaces 2 5.1 Eigenvalues and Eigenvectors Eigenvalue and Eigenvector If A is a n n

### Quantum Computing Lecture 2. Review of Linear Algebra

Quantum Computing Lecture 2 Review of Linear Algebra Maris Ozols Linear algebra States of a quantum system form a vector space and their transformations are described by linear operators Vector spaces

### 4. Linear transformations as a vector space 17

4 Linear transformations as a vector space 17 d) 1 2 0 0 1 2 0 0 1 0 0 0 1 2 3 4 32 Let a linear transformation in R 2 be the reflection in the line = x 2 Find its matrix 33 For each linear transformation

### Linear Algebra Highlights

Linear Algebra Highlights Chapter 1 A linear equation in n variables is of the form a 1 x 1 + a 2 x 2 + + a n x n. We can have m equations in n variables, a system of linear equations, which we want to

### MATRICES. knowledge on matrices Knowledge on matrix operations. Matrix as a tool of solving linear equations with two or three unknowns.

MATRICES After studying this chapter you will acquire the skills in knowledge on matrices Knowledge on matrix operations. Matrix as a tool of solving linear equations with two or three unknowns. List of