Math for Machine Learning: Gaussian Elimination | #mathmachinelearning | #linearalgebra | #dogmathc

#LinearAlgebra #MathForML #MachineLearning #GaussianElimination #MLMath #GATEMaths #dogmathic How Do You Write the Full Solution to Ax = b? | MML Exercise 2.6 Gaussian elimination and free variables show up together in this one, and the goal is not just to solve the system but to write the complete solution as a parametric vector. Exercise 2.6 from Mathematics for Machine Learning gives a system Ax = b with six unknowns and asks for all solutions. The augmented matrix goes up first. Before any row operations, two columns are already all zeros. That means x1 and x3 are free variables before a single operation is performed. Two row operations later, the matrix is in row echelon form. The staircase appears. Pivots land on x2, x4, and x5. The remaining columns, x1, x3, and x6, are free. We call them r, s, and t, ranging over all real numbers. Back-substitution takes it from there. Equation three gives x5 in terms of t. That feeds into equation one to get x2. Equation two gives x4. Once every variable is expressed in terms of r, s, and t, the solution gets stacked into a single vector. Constants go in one column vector, then the r terms, the s terms, the t terms. Four vectors total. That is the complete solution set. The parametric vector form is worth learning to write cleanly. It shows up constantly in linear algebra and it makes the structure of the solution visible in a way that a list of equations does not. Topics covered: Gaussian elimination, row echelon form, free variables, pivot variables, augmented matrix, parametric solution, back-substitution, systems of linear equations, Ax = b, row operations, linear algebra, solution set, vector form, MML exercise, mathematics for machine learning Support Dogmathic https://ko-fi.com/dogmathic https://dogmathic.com/ matherssen(at)gmail.com    • Math for Machine Learning: Matrix Row Ops ...      • Mathematics for Machine Learning      • Linear Algebra      • Group Theory      • Abstract Algebra      • Python Math Animations Using Manim   From the book: Mathematics for Machine Learning Marc Peter Deisenroth, A. Aldo Faisal, Cheng Soon Ong Cambridge University Press Book website: https://mml-book.github.io/ This video is part of the Mathematics for Machine Learning playlist. Properties and Concepts Used: Gaussian elimination (row reduction to row echelon form) Gauss-Jordan elimination (reduced row echelon form) Augmented matrix construction Row operations (swap, scale, add/subtract multiples) Pivot variables vs free variables Zero columns as immediate free variable indicators Back-substitution Parametric solution form Parametric vector decomposition (constant + r-vector + s-vector + t-vector) Systems of linear equations Ax = b Solution set notation Staircase / echelon structure Consistency of a linear system Chapters: 0:00 Introduction and setup 1:37 The system and what we are solving 1:55 Gaussian vs Gauss-Jordan elimination 3:14 Building the augmented matrix 4:41 Free variables spotted from zero columns 5:26 Row operations 8:06 Identifying pivots and free variables 9:36 Writing the three equations from row echelon form 10:26 Back-substitution 13:06 Cleaning up and listing all variables 13:56 Parametric vector form of the solution 16:29 Final solution set #Dogmathic #LinearAlgebra #MathForML #MachineLearning #GaussianElimination #MLMath #GATEMaths