Explore the geometric interpretation of linear equation systems in this comprehensive lecture from the Wild Linear Algebra series. Delve into the association between m equations in n variables, m by n matrices, and linear transformations. Gain insights into solution existence and characteristics through kernel and rank concepts. Discover a novel approach to subspaces using properties rather than infinite sets, aligning with modern computer science. Examine illustrative examples covering various dimensions, linear transformations, and subspace properties. Learn about kernel and image properties, hyperplanes, and spanning sets. Engage with theorems, proofs, and exercises to solidify understanding of these fundamental linear algebra concepts.
The Geometry of a System of Linear Equations - Wild Linear Algebra A
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Overview
Syllabus
CONTENT SUMMARY: pg 1: @ The geometry of a system of linear equations; m linear equations in n variables;
pg 2: @ The picture to keep in mind; The big picture;
pg 3: @05:00 The kernel property; property versus set; remark on fundamental issue @ see "math foundations" series;
pg 4: @ examples; what is a line?; what is a circle; properties instead of infinite sets;
pg 5: @ managing properties; statement of Properties moral;
pg 6: @ examples; properties of a 3-d vector;
pg 7: @ Subspace properties; Definition and examples;
pg 8: @ subspace properties of 2-d vectors;
pg 9: @ subspace properties of 3-d vectors;
pg 10: @ Definition of kernel property; definition of image property; Theorem 1; Theorem 2;
pg 11: @ Theorem proofs;
pg 12: @32:05 subspaces in higher dimensional spaces; spanning set; equation set; hyperplane @ ;
pg 13: @ Linear transformation n-dim to m-dim; pg13_Theorem ;
pg 14: @ proof of pg13_Theorem;
pg 15: @ example 2d to 2d;
pg 16: @ example 3d to 2d;
pg 17: @ example 3d to 3d;
pg 18: @1:03:13 example continued; remark: typifies a linear transformation @;
pg 19: @ exercise 18.1;
pg 20: @ exercise 18.2; THANKS to EmptySpaceEnterprise
Introduction
The kernel property
What is a line? What is a circle;
Managing properties
Properties of vectors
Subspace properties
Subspaces of V²
Subspaces of V³
Definition of kernel property
subspaces in higher dimensional spaces
Linear transformation n-dim to m-dim
Taught by
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