Explore the applications of 2x2 matrices in this comprehensive 44-minute lecture from the Wild Linear Algebra series. Delve into the fundamental problem of inverting coordinate changes and its applications in solving systems of two linear equations. Examine the dual interpretations of this problem in terms of line intersections and vector combinations. Learn about the algebraic introduction of vectors and matrices, and discover how composing two variable changes leads to the basic matrix multiplication formula. Investigate the close connection between matrix multiplication and determinants. Cover topics including change of coordinates, vector interpretation of linear systems, matrix notation, arithmetic with matrices and vectors, laws of vector arithmetic, geometric interpretations of abstract column vectors, matrix operations, and determinants. Engage with practical examples and exercises to reinforce your understanding of these crucial linear algebra concepts.
Overview
Syllabus
CONTENT SUMMARY: pg 1: @ inverting the relationship between two pairs of variables;
pg 2: @ application of change of coordinates;
pg 3: @ changing coordinates is related to solving a system of linear equations; a family of problems
pg 4: @ example with parallel lines; zero determinant then 2 lines parallel 2-dim;
pg 5: @ Vector interpretation of a linear system;
pg 6: @ example of 2 vector system without solution;
pg 7: @ change of coordinates as heart of the subject;
pg 8: @ generalize the example on the previous page;
pg 9: @ matrix notation; a column vector; matrix/vector multiplication;
pg 10: @ writing a pair of linear equations in matrix/vector form;
pg 11: @ arithmetic with matrices and vectors; introducing notation independent of application; you might think of this page as the start of a course in linear algebra; column vectors, scalar multiplication, addition, subtraction; approach is independent of any geometric interpretation;
pg 12: @ laws of vector arithmetic;
pg 13: @ geometrical interpretation of abstract column vectors;
pg 14: @ A matrix as an array of numbers; scalar multiplication, addition, subtraction; matrices follow the same laws as vectors;
pg 15: @ define product of matrix and a column vector; define product of two 2by2 matrices;
pg 16: @ examples;
pg 17: @ determinants; alternate notation; theorem: determinant of product of matrices is equal to the product of the determinants of 2 matrices; exercises 5.1:2 ;
pg 18: @ exercises 5.3:5 ; THANKS to EmptySpace Enterprise
Introduction
Arithmetic with 1×1 matrices / 2×2 matrices
Vector interpretation of a linear system
2×2 matrices
change of coordinates
Example. ABC=ABC
matrix notation; a column vector; matrix/vector multiplication
Four important matrices
determinants
Taught by
Insights into Mathematics
