Linear Algebra

Eigenvalues and Eigenvectors – Linear Algebra

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What actually are eigenvalues and eigenvectors? For any square matrix $A$, an Eigenvector ($v$) and Eigenvalue ($\lambda$) satisfy the linear algebra equation. In other words, when $A$ is multiplied by the eigenvector ($v$), the resulting vector points in the exact same direction (or opposite), scaled only by the scaler factor $\lambda$. How to find eigenvalue? ... Read More

Markov Chains – Linear Algebra

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Understanding Markov Chains: The Memoryless Math In probability and statistics, some systems look backward to determine the future. Whereas Markov Chains look only at the present moment. It was named after the Russian mathematician Andrey Markov. These mathematical models describe systems that jump between different states over time. The Markov Property: The future state of ... Read More

Linear Algebra, Optimization Problem

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Linear algebra is used to solve various optimization problems. Basic concepts in Linear algebra include vectors, vector space, linear transformations, systems of linear equations. While optimization problems can be linear or non linear. Vectors: Reference:C. C. Aggarwal, Linear Algebra and Optimization for Machine Learning: A Textbook. Cham: Springer International Publishing, 2020. doi: 10.1007/978-3-030-40344-7. Y. Geophys, ... Read More

Matrix Transformations – Linear Algebra

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1. Functions and Transformations In standard algebra, a function takes a single number as an input and outputs a new number: f(x) = y. In Linear Algebra, we scale this concept up. A Transformation (often denoted as T) is a mathematical rule that takes an entire Vector as an input and assigns it to a ... Read More

Algebraic Properties of Matrices

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1. Properties of Matrix Arithmetic Matrix arithmetic shares many rules with standard algebra, but there are crucial differences. Here are the core laws: Commutative Law for Addition: A + B = B + A Associative Law: A + (B + C) = (A + B) + C A(BC) = (AB)C Distributive Law: A(B + C) ... Read More

Matrices and Matrix Operations – Linear Algebra

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1 & 2. Matrix Notation, Terminology & Equality A Matrix is simply a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. Matrices are considered Equal only if they have the exact same dimensions AND every corresponding element is identical. 2 4 -1 9 = 2 4 -1 9 Conversely, here is ... Read More

Inner or Dot Products – Linear Algebra

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In linear algebra, the dot product (or inner product) is specifically used for vectors. It takes two vectors of the same dimension and results in a single scalar value (a number). 1. Geometric Definition Geometrically, the dot product measures the overlap or projection of one vector onto another. a · b = ||a|| ||b|| cos(θ) ... Read More

Systems of Linear Equations and Matrices

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References: Linear Equation: At its simplest, a linear equation is an algebraic equation that creates a straight line when plotted on a graph. Every variable in the equation is raised to the first power (meaning no exponents like $x^2$ or $y^3$), and there are no variables multiplied by each other. The Standard Forms Depending on ... Read More

Linear Algebra – Gaussian Elimination

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Book: Introduction to Linear Algebra, Johnson et al. Third Edition Gaussian Elimination is a method in linear algebra used to solve systems of linear equations.It systematically transforms a system of equations into a simpler form so the solutions become easy to find. It works by manipulating the augmented matrix of the system using basic row ... Read More

Rank of Matrix

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The Rank of a Matrix is a fundamental concept in linear algebra that tells you how much “information” or “dimensionality” a matrix preserves. At its core, the rank represents the number of linearly independent rows or columns in a matrix. It is the dimension of the vector space generated by its columns (or rows). https://www.mathsisfun.com/algebra/matrix-rank.html ... Read More
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