Linear Algebra BAS113 Lecture 2 Introduction

Course Information

• Course name: Linear Algebra (BAS113)
• University: Sphinx University
• Instructors: Dr. Amira A. Allam and Dr. Mahmoud Owais

Lecture 2: Introduction

• Lectures notes are available, and the white board is crucial for the course.
• Take notes during the lectures by copying the white board.
• Complete all homework problems.
• Review the material before each exam by referencing the textbook, lecture notes, and model homework answers.

Numbers

• Real numbers comprise rational and irrational numbers.
• Rational numbers include integers, whole numbers, and natural numbers; examples provided: 5/6, 1.5, 3.78, 6/9
• Irrational numbers include numbers like √5 and -√2.
• Integers include positive and negative whole numbers and zero; examples provided: 3, 4, -8, -3

Whole Numbers

• Whole numbers are non-negative integers; example: {0, 1, 2, 3...}
• Natural numbers are positive whole numbers; example: {1, 2, 3...}

Algebra

• Algebra uses letters (variables) to represent unknown numbers.
• Known numbers in an algebraic expression are called constants.
• Includes basic math operations (addition, subtraction, multiplication, and division) with both variables and constants.
• Example: x + 10 = 0

Laws in Algebra

• Commutative Law: Changes the order of the terms in addition or multiplication; Examples: a + b = b + a, a * b = b * a
• Associative Law: Changes the grouping of terms in addition or multiplication; Example: (a + b) + c = a + (b + c)
• Distributive Law: Combines addition and multiplication; Example: a (b +c) = a(b) + a(c)

Sets

• A set is a collection of objects (elements or members).
• Sets can contain any kind of objects, even other sets.
• Capital letters represent sets, lowercase letters represent set elements.
• A is a subset of B (A⊂B) if every element in A is also in B.
• Set notations are used to define sets; example: N = the set of natural numbers Z = the set of integers Q =the set of rational numbers R = the set of real numbers

Set-Builder Notation

• Defines sets based on properties or conditions.
• General format: {membership | properties}, example, N = {x ∈ Z | x > 0}.

Functions

• A function assigns each element in one set (domain) to a unique element in another set (codomain).
• Functions are also known as maps.
• Example: f: {a, b, c} to {1, 3, 5, 9} defined by f(a) = 1, f(b) = 5, f(c) = 9.

Complex Numbers

• Complex numbers are ordered pairs of real numbers (a, b).
• 'a' is the real part (Re z), 'b' is the imaginary part (Im z).
• z is purely imaginary if Re z = 0.
• z is real if Im z = 0.
• C represents the set of all complex numbers.
• Operations of addition and multiplication are defined for complex numbers.
• Example: (a,b) + (c,d) = (a + c, b + d), (a,b) (c,d) = (ac − bd, ad + bc)

Conjugate of Complex Numbers

• The conjugate of a complex number (x, y) or (x + iy) is (x, -y) or (x − iy).
• Geometrically, the conjugate is the reflection of the number across the real axis.

Exercises

• Various exercises throughout the presentation, including finding the imaginary part of a complex number, solving quadratic equations, and solving polynomial equations.

This quiz covers the concepts introduced in Lecture 2 of the Linear Algebra course at Sphinx University. Focus on understanding real numbers, rational and irrational numbers, as well as whole and natural numbers. Review material thoroughly to prepare for exams.