Vector Space Let \(V\) be a set with cardinality at least 2 and let \((V,+)\) be an Abelian group with respect to the operation \(+\). Let \(\mathbb{R}\) be the field of real numbers. For any \(c\in\mathbb{R}\) and \(v\in V\), define \(c\cdot v = 0\), where \(0\) is the identity element in the Abelian group \((V,+)\). […]
(a) Find the initial point of the vector that is equivalent to u = (1, 2) and whose terminal point is B(2, 0). (b) Find the terminal point of the vector that is equivalent to u = (1, 1, 3) and whose initial point is A(0, 2, 0). Part (a): Find the initial point of
Question 1 (a) Find the terminal point of the vector that is equivalent to u = (1, 2) and whose initial point is A(1, 1). (b) Find the initial point of the vector that is equivalent to u = (1, 1, 3) and whose terminal point is B(−1, −1, 2). Question2 a) Find the initial
Question 1 (a) Find the terminal point of the vector that is equivalent to u = (1, 2) and whose initial point is A(1, 1). (b) Find the initial point of the vector that is equivalent to u = (1, 1, 3) and whose terminal point is B(−1, −1, 2). Question 2 (a) Find the
(a) Find the terminal point of the vector that is equivalent to u = (1, 2) and whose initial point is A(1, 1). (b) Find the initial point of the vector that is equivalent to u = (1, 1, 3) and whose terminal point is B(−1, −1, 2). (a) Terminal point of the equivalent vector
Why do we need two number systems, and how do they interlock? Real numbers (ℝ) model the world’s continuity, but complex numbers (ℂ) solve its hidden puzzles—like quantum waves and AC circuits. Real numbers It is (denoted by ℝ) and is a set of all rational and irrational numbers, forming a complete, ordered, continuous number line.
Vector Space • A vector space is a fundamental concept in mathematics and physics, particularly in linear algebra.• It is a collection of objects called vectors, which can be added together and multiplied by scalars (real or complex numbers) and they remain objects of the same type. • Vector spaces provide a framework for studying
What is a Subset? A collection of elements is known as a subset of all the elements of the set contained inside another set Example: Set A has {X, Y} Set B has {X, Y, Z}, Then A is the subset of B because elements of A are also present in set B Subset Symbol In
What is Set theory? Set theory is the mathematical theory of well-determined collections, called sets of objects that are called members or elements of the set Elements in a set can be finite or infinite Set theory – Finite set Objects/ Elements of Finite set are – Days of the week: { Sun, Mon, Tue, Wed, Thus, Fri,
What is a Perfect number? A perfect number is a positive integer that is equal to the sum of its proper divisors (factors) excluding itself is called a perfect number. The smallest perfect number is 6, which is the sum of 1, 2, and 3. Other perfect numbers are 28, 496, and 8128. 6 =
What is a Complex number? Complex numbers allow solutions to certain equations that have no solutions in real numbers. A set of complex numbers is denoted by C Example : (x – 1)2 = – 9 x = 1 +√-9 x = 1 + 3i A complex number is a number that can be expressed
What is the Real number? Real numbers are the numbers which include both rational and irrational numbers. It is denoted by R Real number Examples Real number Properties Any non-zero real number is either negative or positive. The real numbers make up an infinite set of numbers The sum of two non-negative real numbers is again a non-negative real number The product of two non-negative
What is an Irrational number? Numbers which are not rational are irrational An irrational number is a number that cannot be described in the form of p/q Numbers that cannot be written as a ratio of two integers are called Irrational It is denoted by P Irrational number Examples *π = 22/ 7, = 3.1428571428571… It
What is a Rational number? A rational number is a number that can be in the form of p / q where p and q are integers and q is not equal to zero. Rational numbers are denoted by Q Rational number Examples Please note that π is not a Rational number *π = 22/ 7, = 3.1428571428571…
What is an Integer number? Integer numbers are the numbers that can be positive, negative, or zero, but cannot be a fraction The integers are generated from : a set of counting numbers 1, 2, 3, . . . And the operation of subtraction. When a larger number is subtracted from a smaller number, the result
Division I have 27 Rupees to distribute to 3 students. How many do you give to each of them? Mathematically 27 ÷ 3 = 9 As there is 0 remainder it can be written as 27 mod 3 =0 29 ÷ 3 = 9 with remainder 2 (29 = 3 x 9 + 2) It
What is a Prime number? Prime numbers are whole numbers greater than 1, that have only two factors 1 and the number itself. Prime numbers are divisible only by the number 1 or itself. Prime number Example For example, 2, 3, 5, 7, and 11 are the first few prime numbers. The factor of 2
Use of Natural number Natural numbers are used for – Counting: For example – there are Four mangoes in the basket Ordering: Example – Indore is the number one city in cleanliness in India Natural number example Number Natural number 0, 1, 2, 3, 4 ……………………………………….………∞ Natural number 13 Natural number 12 Natural number 11
