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Questions and Answers
What are the values known in a given expression called?
What are the values known in a given expression called?
Which law allows changing the order of terms in addition and multiplication?
Which law allows changing the order of terms in addition and multiplication?
In the equation $x + 10 = 0$, what does 'x' represent?
In the equation $x + 10 = 0$, what does 'x' represent?
What does the associative law allow in terms of operations?
What does the associative law allow in terms of operations?
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What is the purpose of the distributive law?
What is the purpose of the distributive law?
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Which operation is NOT involved in algebra as described?
Which operation is NOT involved in algebra as described?
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What does the associative law exemplify when adding the numbers?
What does the associative law exemplify when adding the numbers?
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If a variable is multiplied by a sum, which law applies to distribute the variable?
If a variable is multiplied by a sum, which law applies to distribute the variable?
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In the expression $a + b + c$, how can the associative law be applied?
In the expression $a + b + c$, how can the associative law be applied?
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What does the commutative law indicate for the operation of multiplication?
What does the commutative law indicate for the operation of multiplication?
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Which statement correctly describes a known value in an algebraic expression?
Which statement correctly describes a known value in an algebraic expression?
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Which of the following illustrates the commutative law in an equation?
Which of the following illustrates the commutative law in an equation?
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Study Notes
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.
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Description
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.
