Generalized Dilations and Eigenvalues - Wild Linear Algebra A - NJ Wildberger

Overview

Explore generalized dilations and eigenvalues in this 56-minute lecture from the Wild Linear Algebra series. Delve into the comparison of different coordinate systems, learning how to change from one basis to another. Examine the concept of similar matrices and generalized dilations, focusing on linear transformations that can be diagonalized. Discover the process of finding eigenvectors and eigenvalues of a matrix, considered one of the most crucial problems in linear algebra. Through practical examples and clear explanations, gain insights into ordered bases, coordinate vectors, change of basis matrices, and how linear transformations appear in different frames of reference. Enhance your understanding of this fundamental topic in mathematics, essential for advanced studies in linear algebra and its applications.

Syllabus

CONTENT SUMMARY: pg 1: @ introducing the 2nd most important problem in linear algebra; 2 frames of reference; desire to compare frames of reference; example using "Bob" and "Rachel" basis vectors;
pg 2: @ example of vector change of basis; going back and forth between "Bob's" and "Rachel's" systems;
pg 3: @ notation to facilitate change of basis conversation; ordered bases, coordinate vectors, change of basis matrix; change of basis matrices are inverse matrices;
pg 4: @ process to obtain change of basis matrix; examples to verify agreement to earlier results;
pg 5: @ how linear transformations appear when going from one frame of reference to another frame of reference; start with an example;
pg 6: @ example continue; the same linear transformation expressed in different frames of reference; the transformation is much more easily expressed in "Rachel's" system;
pg 7: @ Definition of similar matrices; Similar matrices represent the same linear transformation but with respect to w.r.t. different bases
Introduction
Between Bob and Rachels systems
Ordered bases and coordinate vectors
Changing bases for a linear transformation
Similar matrices
Generalized dilations