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15) Vector Norms

Vector Norms Vector Norms: Measuring Vector Magnitude 1. What is a Vector Norm? A vector norm is a function that assigns a positive length to a vector, measuring its size. Real-World Meaning Physics: The norm of velocity gives the speed of an object. Machine Learning: Norms measure distances between data points. Optimization: Used in gradient descent to control updates. 2. Types of Vector Norms (a) Euclidean Norm (L2 Norm) Measures straight-line distance. ||v||₂ = sqrt(v₁² + v₂² + ... + vₙ²) Real-World Use: GPS Navigation: Finding shortest distance. Physics: Computing magnitudes of forces. (b) Manhattan Norm (L1 Norm) Sum of absolute values of vector components. ||v||₁ = |v₁| + |v₂| + ... + |vₙ| Real-Worl...
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14) Cramer's Rule

Cramer's Rule - Solving Systems of Equations Cramer's Rule - Solving Systems of Linear Equations 1. Understanding Cramer's Rule Cramer’s Rule is used to solve systems of n linear equations with n unknowns using determinants. It applies only when the determinant of the coefficient matrix is nonzero. 2. Step-by-Step Algorithm Step 1: Compute det(A) Find the determinant of matrix A . If det(A) = 0 , the system has no unique solution. Step 2: Construct Matrices A_1, A_2, ... For each variable, replace the corresponding column of A with the right-hand side values ( B ). Step 3: Compute Determinants Find det(A_1), det(A_2), ... for each modified matrix. Step 4: Compute Solutions Use the formula: x_i = det(A_i) / det(A) 3. Real-World Applications Com...
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13) Determinants and their applications

Determinants & Their Applications Determinants & Their Applications 1. What is a Determinant? A determinant is a scalar value computed from a square matrix. It encodes important information about the matrix and the linear transformation it represents. Essentially, the determinant tells us: Volume Scaling Factor: How much the transformation scales area (2D), volume (3D), or higher-dimensional “volume.” Invertibility: A matrix is invertible if and only if its determinant is non-zero. Real-World Meaning: When you apply a transformation like rotating, scaling, or shearing to a shape, the determinant tells you how the area or volume changes. For example, a determinant of 2 means the area is doubled. In physical systems, a non-zero determinant ensures that a system of equations (e.g., forces in equilibrium) has a unique solution. 2. How to Compute...
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12) Markov Chain

Markov Chains Explained Markov Chains Explained 1. What is a Markov Chain? A Markov Chain is a mathematical system that undergoes transitions from one state to another based on certain probabilities. Its key property, known as the Markov Property , is that the future state depends only on the current state, not on the sequence of events that preceded it. Real-World Examples: Weather Prediction: Tomorrow’s weather depends only on today’s, not on the entire past week. Stock Market Trends: A stock's price tomorrow depends on today's market condition. Google PageRank: The probability of moving from one webpage to another depends on the current browsing behavior. 2. Components of a Markov Chain A Markov Chain consists of the following key components: States: The possible conditions or positions of the sy...
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11) Principal Component Analysis

Principal Component Analysis (PCA) Principal Component Analysis (PCA) 1. Why Do We Need PCA? Problem: High-dimensional data is hard to visualize and process. Some dimensions contain redundant or irrelevant information. Real-World Example: In face recognition, a raw image has thousands of pixels, but PCA helps extract the most informative patterns. 2. What PCA Does PCA transforms data into a new coordinate system where the first principal component captures the most variance. Real-World Example: In stock market analysis, PCA finds underlying trends affecting multiple stocks. 3. Steps of PCA Step 1: Standardize the Data Formula: X_scaled = (X - μ) / σ Example: Standardizing height and weight ensures fair comparison. Step 2...
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10) Single Value Decomposition

Singular Value Decomposition (SVD) Singular Value Decomposition (SVD) 1. What is SVD? Singular Value Decomposition (SVD) is a matrix factorization technique that breaks down a matrix into three special matrices: A = U Σ Vᵀ U : Orthogonal matrix (left singular vectors) Σ : Diagonal matrix (singular values) Vᵀ : Orthogonal matrix (right singular vectors) Real-World Meaning: SVD extracts important features from data and is widely used in machine learning, image processing, and recommendation systems. 2. Structure of SVD Components If A is an (m × n) matrix: U (m × m): Represents variance in row space. Σ (m × n): Contains singular values (importance of components). Vᵀ (n × n): Represents variance in column space. Real-World Meaning: Like breaking down an im...
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9) Diagonalization

Diagonalization Explained Diagonalization: Breaking Matrices into Simpler Forms 1. What is Diagonalization? Diagonalization is the process of converting a square matrix A into a diagonal matrix D using a transformation matrix P such that: A = P D P -1 2. Why is Diagonalization Useful? Simplifies Matrix Operations: Makes computing powers of A easier. Speeds Up Computations: Important in physics, AI, and finance. Uncovers Hidden Structures: Helps in dimensionality reduction. 3. Steps to Diagonalize a Matrix Find Eigenvalues: Solve det(A - λI) = 0. Find Eigenvectors: Solve (A - λI)x = 0. Form Matrices P and D: P contains eigenvectors, D contains eigenvalues. 4. Conditions for Diagonalizability A matrix is diagonalizable if it has n linearly independent eigenvectors. If eigenvectors are dependent, the matrix is not diagonalizable. ...
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