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WildLinAlg17: Rank and Nullity of a Linear Transformation

This is a full hour lecture in which we step up to linear transformations with spaces of more than 3 dimensions, introduce the kernel and the image properties, and the corresponding dimension numbers called nullity and rank.Most of the lecture looks in detail at a particular transformation from four to three dimensional space. We discuss how to visualize four dimensions in a way that is consistent with our pictures of two and three dimensions.The main computations rest on our understanding of row reduction of a matrix.CONTENT SUMMARY: pg 1: @00:08 Lesson about nullity and rank of a linear transformation; kernel and image of linear transformation; general linear transformations; Example? (mxn is 3x4);pg 2: @03:16 How to visualize in higher dimensions; shift from affine space to vector space; points/vectors;pg 3: @09:17 4-dimensional space (algebraically);pg 4: @11:00 4-dimensional space (geometrically);pg 5: @16:52 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: @23:05 Definition of kernel vector; kernel property;pg 7: @24:15 Finding vectors with the kernel property for a transformation using row reduction;pg 8: @28:32 Definition of image vector; image property; pg 9: @30:17 at least the columns of the transformation matrix have this image property; pg 10: @32:57 Finding vectors with the image property for a transformation using row reduction; pg 11: @37:11 Another approach to the image of a transformation; the column space of a matrix; pg 12: @40:37 The whole picture; kernel,image, nullity, rank; pg 13: @43:21 Important observations;?pg 14: @45:51 relationship between the nullity and the rank; Rank-Nullity theorem; pg 15: @48:27 example: Linear transformation from 3dim space to 4dim space; kernel, image, rank, nullity; pg 16: @50:12 example continued; kernel; pg 17: @52:43 example continued; image; remark on relationship of columns in row reduction @53:07; pg 18:? @55:05 example summary; pg 19: @58:21 exercises 17.(1:2); kernel property, image property; pg 20: @59:03 exercise 17.3 ; describe ker() and im(); (THANKS to EmptySpaceEnterprise)
Length: 01:01:09


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