Linear Algebra Essentials: Matrix Theory With Applications

Master rank, systems of equations, eigenvalues, Cayley-Hamilton theorem, and diagonalization with step-by-step examples.

What you'll learn

Understand the rank of a matrix and its significance.

Solve systems of linear equations using matrix methods.

Compute and interpret eigenvalues and eigenvectors.

Apply the Cayley-Hamilton Theorem to simplify matrix computations.

Perform diagonalization and use it to solve problems.

Strengthen your foundation in Linear Algebra with applications to engineering, data science, and computer science.

Requirements

Basic knowledge of algebra.

Curiosity to learn mathematics with applications

Description

Matrix Theory forms the backbone of Linear Algebra, serving as a powerful mathematical tool with applications spanning Engineering, Physics, Computer Science, Data Science, and Machine Learning. A strong understanding of matrices is essential for solving real-world problems, from modeling physical systems to building intelligent algorithms.This course is designed to provide you with a solid foundation in matrix concepts, gradually building your knowledge and guiding you through both fundamental and advanced ideas. We will begin with the basics such as the rank of a matrix and the solution of systems of linear equations, ensuring you develop a clear conceptual understanding. From there, we will move on to more advanced topics including eigenvalues and eigenvectors, which are critical in areas such as stability analysis, principal component analysis, and quantum mechanics.The course also covers powerful theorems like the Cayley-Hamilton Theorem, and techniques such as Matrix Diagonalization, which simplify complex computations and deepen your insight into linear transformations.Each concept is explained in a step-by-step manner, supported by intuitive explanations, worked examples, and practice exercises. By the end of the course, you will gain confidence in applying matrix theory to mathematical modeling, computational methods, and advanced problem-solving, strengthening both your academic and professional journey.

Overview

Section 1: Introduction

Lecture 1 Introduction to Matrix Theory

Lecture 2 Introduction to Complex Matrices

Lecture 3 Rank of a Matrix - Echelon Form

Lecture 4 Solved Examples on Rank by Echelon form

Lecture 5 Rank of a Matrix - Normal Form

Section 2: Systems of Linear Equations

Lecture 6 Solutions of System of Equations (Part 1)

Lecture 7 Solutions of System of Equations (Part 2)

Lecture 8 Homogeneous System of Equations (Part 1)

Lecture 9 Homogeneous System of Equations (Part 2)

Section 3: Eigenvalues and Eigenvectors

Lecture 10 Introduction to Eigen Values and Eigen Vectors

Lecture 11 How to use calculator for Matrix Operations - Telugu (Full and Clear Explanation

Lecture 12 Evaluating Eigen Values and Eigen Vectors (TELUGU)

Lecture 13 Additional Problems on Eigen values and Eigen Vectors

Lecture 14 Properties of Eigen Values

Section 4: Cayley-Hamilton Theorem

Lecture 15 Cayley Hamilton Theorem_ Part - 1

Lecture 16 Cayley Hamilton Theorem Part-II

Section 5: Diagonalization

Lecture 17 Diagonalization of a Matrix Part - 1

Lecture 18 Diagonalization of a Matrix_ Part II

Undergraduate & postgraduate students studying Mathematics, Engineering, or Computer Science.,Researchers who need matrix methods for applied problems.,Students preparing for exams like GATE, GRE, CSIR-NET, IIT-JAM, or university courses.,Professionals in data science, machine learning, and computational fields looking to strengthen their math foundations.