WildLinAlg12: Generalized dilations and eigenvalues
This is 12th lecture in this course on Linear Algebra by N J Wildberger. Linear algebra is all about different perspectives. Here we compare Bob and Rachel's coordinate system and learn how to change from one basis to another. Then we define similar matrices and generalized dilations---those linear transformations that are similar to a (mixed) dilation, or can be diagonalized. We look at an example to sketch the basic idea.This leads to the second most important problem in the subject: how to find the eigenvectors and eigenvalues of a matrix.CONTENT SUMMARY: pg 1: @00:08 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: @06:27 example of vector change of basis; going back and forth between "Bob's" and "Rachel's" systems;?pg 3: @09:34 notation to facilitate change of basis conversation; ordered bases, coordinate vectors, change? of basis matrix; change of basis matrices are inverse matrices;pg 4: @14:11 process to obtain change of basis matrix; examples to verify agreement to earlier results;pg 5: @16:47 how linear transformations appear when going from one frame of reference to another frame of reference; start with an example;pg 6: @22:29 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: @26:20 Definition of similar matrices; Similar matrices represent the same linear transformation but with respect to (w.r.t.) different bases; example; the similarity relation is symmetric;pg 12: @38:45 definitions for eigenvector and eigenvalue; example;pg 13: @42:54 example of finding eigenvector's and associated eigenvalue's in the 2x2 case; the characteristic equation;pg 14: @48:15 example continued; the eigenvalues make the associated matrix not-invertible; the previous as a fundamental derivation;pg 15: @51:56 exercise 12.1;pg 16: @53:14 exercises 12.(2:4) ; remark on a problem? encountered in the previous solution @54:00; (THANKS to EmptySpaceEnterprise)
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