Vectors in \(\mathbb{R}^n\) and \(\mathbb{C}^n\) Introduction to Vectors A vector is a mathematical object that has both magnitude and direction. Vectors are essential in physics, engineering, and mathematics. They can be represented in different dimensions, such as real number space \(\mathbb{R}^n\) and complex number space \(\mathbb{C}^n\). Visualization of Vectors in \(\mathbb{R}^3\) Consider a three-dimensional space […]
Introduction to Vectors Read More »
Some Questions of Linear Algebra: Linear Transformation Definition: Linear Transformation A linear transformation \(T: \mathbb{R}^n \to \mathbb{R}^m\) is a function that satisfies: Additivity: \(T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})\) for all \(\mathbf{u}, \mathbf{v} \in \mathbb{R}^n\). Homogeneity: \(T(c \mathbf{v}) = c T(\mathbf{v})\) for all \(\mathbf{v} \in \mathbb{R}^n\) and scalars \(c\). Numerical Example: Linear Transformation
Some Questions of Linear Algebra: Linear Transformation Read More »
Addition Law of Probability and Examples Data Science and A.I. Lecture Series Introduction to Addition Law Statement: For any two events \( A \) and \( B \), the probability of their union is given by: \[ P(A \cup B) = P(A) + P(B) – P(A \cap B) \] This formula calculates the probability
Addition Law of Probability and Examples Read More »
Set Operations and Important Laws Data Science and A.I. Lecture Series Set Operations Union Definition: \[ A \cup B = \{x : x \in A \text{ or } x \in B\} \] Examples: \( A = \{1, 2, 3\}, B = \{3, 4, 5\} \implies A \cup B = \{1, 2, 3, 4, 5\}
Set Operations and Important Laws Read More »
Venn Diagrams – Data Science and AI Lecture Series Welcome to our Data Science and AI Lecture Series! In this post, we’ll dive into the world of Venn Diagrams, an essential tool in set theory that simplifies understanding the relationships between sets. Whether you’re studying mathematics, data science, or AI, mastering concepts like intersections, unions,
& Hierarchy of Sets Universal Set (\( U \)) The universal set (\( U \)) contains all elements under consideration in a specific context. Example 1: If studying numbers, \( U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \). Example 2: If studying letters, \( U = \{\text{a, b, c, …,
Understanding Permutations Data Science and A.I. Lecture Series Author: Bindeshwar Singh Kushwaha PostNetwork Academy Introduction to Permutations A permutation is an arrangement of objects in a specific order. The order of arrangement is crucial in permutations. Example: Arranging the letters of the word “ABC”. Total permutations = $3! = 6$. Key Formula for Permutations The
Understanding Permutations Read More »
Group in Algebra Let G be a non-empty set and * (multiplication) be a binary operation defined on G. Then algebraic structure (G, *) is called group if it satisfies the following axioms. 1- Closure property If a∈G, and b∈G then a*b∈G it is called closure property. of G 2- Associativity If a∈G, b∈G ,