Linear Algebra: Systems, Matrices & Transformations

This presentation explores Richard Feynman's geometric demonstration of planetary elliptical orbits under an inverse-square law, a proof notable for its elegance and minimal reliance on calculus.

This lecture introduces fundamental concepts of linear systems, covering their definition, methods of representation including matrix and vector forms, and the elementary operations essential for solving them. Emphasis is placed on the geometric interpretation of solutions.

This lecture introduces fundamental concepts of linear algebra, including vector operations, matrix definitions, and the practical application of Gauss-Jordan elimination for solving systems of linear equations.

This lecture introduces the fundamental concepts of linear systems, focusing on the definition and calculation of matrix rank, and formalizes matrix algebra operations including matrix-vector products.

This session provides a structured walk-through of foundational linear algebra problems, demonstrating techniques for solving linear systems, achieving Reduced Row Echelon Form, and computing the rank of a matrix.

This lecture introduces the fundamental concept of linear transformations, defining their properties and demonstrating how they map vectors between vector spaces. It covers the algebraic criteria for linearity and provides examples of geometric transformations.

This lecture provides a foundational introduction to geometric linear transformations, detailing orthogonal projection, reflection, scaling, rotation, and shearing. It emphasizes their algebraic representation via matrices and their visual interpretation in Euclidean space.

This session works through additional problems concerning the matrix-vector product Ax and fundamental properties of linear transformations, reinforcing concepts critical for understanding vector space mappings.

This lecture provides a rigorous introduction to the matrix product, detailing its definition, algebraic properties, and geometric interpretations within linear transformations. Key topics include associativity, distributivity, and non-commutativity.

This lecture provides a comprehensive treatment of the inverse of a linear transformation, covering definitions, properties, and methods for computation. It explores the relationship between invertibility and bijectivity for linear maps.