Free online courseLinear Algebra Course

Course Description

Welcome to the Linear Algebra Course, a comprehensive foundational course with a duration of 10 hours and 49 minutes. As part of the Basic Studies category and specifically focusing on Algebra, this course provides an in-depth exploration of critical concepts and techniques in Linear Algebra. Whether you are a student seeking a solid grounding in the subject or looking to refresh your knowledge, this course is tailored to meet your needs.

The course starts by delving into Systems of Linear Equations, where you'll learn to solve these systems using Augmented Matrices and Row Operations. This sets the stage for understanding the intricacies of Row Reduction and Echelon Forms. As you progress, you'll become adept at interpreting Solution Sets and identifying Free Variables.

Next, we move on to Vector Equations and Linear Combinations, essential tools for handling more complex algebraic structures. A thorough understanding of the Matrix Equation Ax=b and the Computation of Ax is also covered, ensuring you are well-equipped to tackle both Homogeneous and Non-Homogeneous System Solutions.

The course also highlights practical applications of linear systems, including Economic Sectors and Network Flow, making the abstract concepts tangible and relevant to real-world scenarios. You'll explore Linear Independence and learn Special Ways to Determine Linear Independence, gaining insights into the robustness of vector spaces.

Matrix Transformations and an Introduction to Linear Transformations form another critical part of the course, offering a gateway to more advanced topics. Matrix Operations such as Sums, Scalar Multiples, Multiplication, and Transpose are discussed in detail. Furthermore, you'll grasp the concepts behind the Inverse of a Matrix, its properties, and methods for solving systems using the inverse, including Elementary Matrices and the Algorithm for finding an inverse.

Characterizations of Invertible Matrices prepare you for the next phase, where we delve into Determinants, including Introduction to Determinants, Co-factor Expansion, and key Properties of Determinants. You'll then advance to Vector Spaces, understanding their structure and the concept of Subspaces.

The exploration continues with Null Spaces, Column Spaces, and the importance of Linearly Independent Sets and Bases, supported by the Spanning Set Theorem. You'll also learn to determine the Dimension of a Vector Space and examine Subspaces of Finite Dimensional Space.

Further, topics like Row Space and Rank provide a well-rounded view of vector operations, setting the stage for Eigenvectors and Eigenvalues and their significance. The course also delves deeper into these concepts, linking them with Determinants and the IMT, and unraveling the Characteristic Equation.

The final module covers Inner Product, Vector Length, and Distance, paving the way to understanding Orthogonal Vectors. You'll encounter Orthogonal Sets and Orthogonal Projections, culminating in the Orthogonal Decomposition Theorem and the Best Approximation Theorem. The study concludes with an examination of Least Squares Problems, ensuring you're proficient in one of the fundamental techniques in numerical methods.

Embrace this opportunity to master Linear Algebra and elevate your algebraic skills to new heights. Enroll now and embark on your journey through the fascinating world of vectors, matrices, and transformations.