Linear Algebra

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

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

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

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

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

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