Change of Coordinates and Determinants - Wild Linear Algebra A

Dive into the fifth lecture of a comprehensive linear algebra course, exploring the fundamental concepts of coordinate changes and determinants. Learn how to invert coordinate transformations and apply this knowledge to solve systems of two linear equations in two unknowns. Discover the dual interpretations of this problem: one involving line intersections and another focusing on vector combinations. Explore the purely algebraic introduction to vectors and matrices, and understand how composing two variable changes leads to the basic matrix multiplication formula. Investigate the close connection between this process and determinants. Cover topics such as vector arithmetic, matrix operations, geometrical interpretations of abstract column vectors, and the properties of determinants. Engage with practical examples and exercises to reinforce your understanding of these crucial linear algebra concepts.

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 coordinants 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
Linear system of equations
Vector interpretation of a linear system
Change of coordinates
Matrix notation
Column vectors and matrices
Laws
Geometrical Interpretations
Matrices
Determinants