Explore the applications of row reduction (Gaussian elimination) in this comprehensive 42-minute lecture on linear algebra. Learn how to tackle the three main problems of linear algebra: inverting linear coordinate changes, computing eigenvalues and eigenvectors, and calculating determinants of square matrices. Follow along with detailed examples and explanations as the lecturer demonstrates each problem-solving technique. Gain insights into the physical meaning of eigenvector equations, the properties of determinants, and the significance of upper triangular matrices. Practice your skills with provided exercises on inverting systems, finding inverse matrices, computing eigenvalues and eigenvectors, and calculating determinants. Enhance your understanding of this fundamental mathematical concept through clear explanations and practical applications.
Applications of Row Reduction - Gaussian Elimination I - Wild Linear Algebra A - NJ Wildberger
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Overview
Syllabus
CONTENT SUMMARY: pg 1: @ 3 main problems of Linear Algebra;
pg 2: @ Inverting a linear change of coordinates; example;
pg 3: @ example finished; new idea: introduce a y-sub-i matrix; to obtain the inverse of a matrix;
pg 4: @ Theorem concerning an invertible matrix;
pg 5: @ Finding eigenvalues and eigenvectors of an nXn matrix; remark about the Homogeneous case;
g 6: @14:23 The eigenvalue problem using row reduction; example1; check of result @;
pg 7: @ example2 as a reminder of the physical meaning of an eigenvector equation see WildLinAlg7;
pg 8: @ example2 continued; finding the eigenvectors using row reduction; perpendicular eigenvectors;
pg 9: @27:52 How to calculate a determinant; characteristics of a determinant; as the volume of a parallelpiped; properties of a determinant necessary to do row reduction @;
pg 10: @ the determinant of an upper triangular matrix;
pg 11: @35:11 example: putting a matrix in upper triangular form to obtain its determinant; remark about this lesson @ ;
pg 12: @ exercises 15.1:2 ; invert some systems using row reduction; find inverse matrices;
pg 13: @ exercises 15.3:4 ; find eigenvalues and eigenvectors; compute determinants; THANKS to EmptySpaceEnterprise
Introduction
Inverting a linear change of co - ods
Inverting a square invertible matrix by row reduction
Finding eigenvalues and eigenvectors
The eigenvalue problem using row reduction
How to calculate a determinant
If A is upper triangular then detA= product of diagonal entries
Exercises: invert some systems using row reduction; find inverse matrices
exercises 15.3:4 ; find eigenvalues and eigenvectors; compute determinants
Taught by
Insights into Mathematics
