MAT133 Course Notes









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216890 MAT133 Course Notes

MAT133

Course Notes

Tyler Holden

Mathematics and Computational Sciences

University of Toronto Mississauga

tyler.holden@utoronto.ca ii

Contents

1 Introduction1

1.1 Sets and notation

1

1.1.1 Operations on Sets

2

1.2 Functions

4

1.2.1 Operations on Functions

6

1.2.2 Symmetries

7

1.2.3 Roots

9

1.2.4 Piecewise Functions

9

1.2.5 Inverse Functions

10

1.3 Polynomials and Rational Functions

12

1.4 Absolute Values

14

1.4.1 The Absolute Value

14

1.4.2 Relation to Intervals

15

1.4.3 Algebra with Inequalities

16

1.5 Exponential Functions

17

1.5.1 Roots

17

1.5.2 Logarithms

19

1.5.3 The Exponential and Logarithmic Functions

20

1.6 Sigma Notation

22

1.6.1 Geometric Series

25

1.7 Exercises

27

2 Financial Mathematics

31

2.1 Compounding Interest

31

2.1.1 The Mathematics of Simple Compounding Interest

32

2.1.2 Annual Percentage Rates

34

2.1.3 Present Value

38

2.1.4 Bonds

40

2.2 Continuous Compounding Interest

42

2.3 Annuities

44

2.3.1 Future Value of an Annuity

44

2.3.2 Present Value of an Annuity

47

2.3.3 Amortization

50

2.4 Perpetuities

53

2.5 Exercises

54
iii

3 Linear Algebra64

3.1 Linear Equations and Systems

64

3.1.1 Number of Solutions

66

3.2 Vectors and Matrices

66

3.2.1 Vectors

68

3.2.2 Matrices

68

3.2.3 Linear Combinations and Matrix Representations

70

3.3 Solving Linear Systems

72

3.3.1 Gaussian Elimination

74

3.3.2 Systems with No Solution

78

3.3.3 Systems with Innitely Many Solutions

78

3.3.4 The Rank of a Matrix

80

3.4 Other Matrix Operations

83

3.4.1 The Transpose of a Matrix

84

3.4.2 Matrix Multiplication

84

3.5 Matrix Inversion

88

3.6 Determinants

92

3.6.1 Denition

92

3.6.2 Properties of the Determinant

95

3.7 Eigenvalues and Eigenvectors

96

3.8 Exercises

98

4 Probability and Counting

105

4.1 Counting

MAT133

Course Notes

Tyler Holden

Mathematics and Computational Sciences

University of Toronto Mississauga

tyler.holden@utoronto.ca ii

Contents

1 Introduction1

1.1 Sets and notation

1

1.1.1 Operations on Sets

2

1.2 Functions

4

1.2.1 Operations on Functions

6

1.2.2 Symmetries

7

1.2.3 Roots

9

1.2.4 Piecewise Functions

9

1.2.5 Inverse Functions

10

1.3 Polynomials and Rational Functions

12

1.4 Absolute Values

14

1.4.1 The Absolute Value

14

1.4.2 Relation to Intervals

15

1.4.3 Algebra with Inequalities

16

1.5 Exponential Functions

17

1.5.1 Roots

17

1.5.2 Logarithms

19

1.5.3 The Exponential and Logarithmic Functions

20

1.6 Sigma Notation

22

1.6.1 Geometric Series

25

1.7 Exercises

27

2 Financial Mathematics

31

2.1 Compounding Interest

31

2.1.1 The Mathematics of Simple Compounding Interest

32

2.1.2 Annual Percentage Rates

34

2.1.3 Present Value

38

2.1.4 Bonds

40

2.2 Continuous Compounding Interest

42

2.3 Annuities

44

2.3.1 Future Value of an Annuity

44

2.3.2 Present Value of an Annuity

47

2.3.3 Amortization

50

2.4 Perpetuities

53

2.5 Exercises

54
iii

3 Linear Algebra64

3.1 Linear Equations and Systems

64

3.1.1 Number of Solutions

66

3.2 Vectors and Matrices

66

3.2.1 Vectors

68

3.2.2 Matrices

68

3.2.3 Linear Combinations and Matrix Representations

70

3.3 Solving Linear Systems

72

3.3.1 Gaussian Elimination

74

3.3.2 Systems with No Solution

78

3.3.3 Systems with Innitely Many Solutions

78

3.3.4 The Rank of a Matrix

80

3.4 Other Matrix Operations

83

3.4.1 The Transpose of a Matrix

84

3.4.2 Matrix Multiplication

84

3.5 Matrix Inversion

88

3.6 Determinants

92

3.6.1 Denition

92

3.6.2 Properties of the Determinant

95

3.7 Eigenvalues and Eigenvectors

96

3.8 Exercises

98

4 Probability and Counting

105

4.1 Counting


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