Rank and Nullity of a Linear Transformation - Wild Linear Algebra A - NJ Wildberger

Overview

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Explore the concepts of rank and nullity in linear transformations through this comprehensive lecture. Delve into linear transformations in spaces with more than three dimensions, focusing on kernel and image properties, and their corresponding dimension numbers. Examine a detailed example of a transformation from four-dimensional to three-dimensional space, learning how to visualize four dimensions consistently with two and three-dimensional representations. Master row reduction techniques for matrices to compute kernels and images of transformations. Understand the relationship between nullity and rank, including the Rank-Nullity theorem. Practice with exercises on kernel and image properties, and describing kernels and images of specific transformations. Gain insights into higher-dimensional linear algebra and its applications in mathematics and related fields.

Syllabus

CONTENT SUMMARY: pg 1: @ Lesson about nullity and rank of a linear transformation; kernel and image of linear transformation; general linear transformations; Example mxn is 3x4;
pg 2: @ How to visualize in higher dimensions; shift from affine space to vector space; points/vectors;
pg 3: @ 4-dimensional space algebraically;
pg 4: @ 4-dimensional space geometrically;
pg 5: @ linear transformation from 4dim to 3dim; Kernel and image of transformation as fundamental; nullity as dimension of the kernel; rank as dimension of the image;
pg 6: @ Definition of kernel vector; kernel property;
pg 7: @ Finding vectors with the kernel property for a transformation using row reduction;
pg 8: @28:32 Definition of image vector; image property; pg 9: @ at least the columns of the transformation matrix have this image property;
pg 10: @ Finding vectors with the image property for a transformation using row reduction;
pg 11: @ Another approach to the image of a transformation; the column space of a matrix;
pg 12: @ The whole picture; kernel,image, nullity, rank;
pg 13: @ Important observations;
pg 14: @ relationship between the nullity and the rank; Rank-Nullity theorem;
pg 15: @ example: Linear transformation from 3dim space to 4dim space; kernel, image, rank, nullity;
pg 16: @ example continued; kernel;
pg 17: @52:43 example continued; image; remark on relationship of columns in row reduction @;
pg 18: @ example summary;
pg 19: @ exercises 17.1:2; kernel property, image property;
pg 20: @ exercise 17.3 ; describe ker and im; THANKS to EmptySpaceEnterprise