Rank and Nullity of a Linear Transformation | Wild Linear Algebra A 17 | NJ Wildberger

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)

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Here are the Insights into Mathematics Playlists:

[ivanbenko]

Dude, you are awesome. I like how you make that short pause after every statement so that listener can process it and I also like how you use simple terminology. This is a trait of a good teacher.

[rolfdoets]

A very key lecture! Thanks again Norman for your very good and clear explanation. This theorem rank-nullity is used to give the RESOLF puzzles a unique solution.

[Mittens]

This was an incredible well defined lecture. Thank you

[fraze]

Thank you Prof ! An Excellent Lecture !

[njwildberger]

That's right, but not all of the vectors can be multiplied by the matrix A. In 17.2 you must solve the equation T(v)=w when w is the given vector, for v. Maybe there is a solution, maybe not.

[samanthatotalyrules]

Great video! Thanks for your explanations I watched the full hour of this and it really helped! But please can you help me with your excercises 17.1 and 17.2?
How do you go about doing them? in 17.1 do you multiply each parts by your matrix A, and see if it gives you the zero vector?

[njwildberger]

There is no separate image and kernel lecture, I have introduced those concepts in this lecture.

[Waranle]

Please more videos... the whole course if you can, appreciate it thnx

[Igdrazil]

Delightfull cristaline video like always Norman. I would just be much more careful in saying "we are LIVING in 3D". This is very tricky, because by repeating this "mantra" we get to think so. And that's a pitty. We can only say that "our volume caracteristics are representable in a volume space 3D". Which is true but somehow trivial. AND THAT'S IT. More over, our LIVING is NOT reduce to our volume caracteristics. In fact we LIVE in thousands of dim, just to parametrise all the movements of our bones. And millions or billions if we include the "continuous" deformations of our flesh, blood vesels... And if we add our chimistry, our smelling, thermodynamics, emotions, thoughts... a monstruous vector space makes our "living nest", with not necessarilly euclidian quadrance, and eventualy more complex structures adding on (symplectic because of conservation of conjugate mecanical parameters, relativistic because of électromagnétisme all over, quantic because of chemestry). In fact we much more look like a mythic "TREE" with millions of roots diving in "INSIDE" and branches communicating with "OUTSIDE", with "OUTSIDE" not being at all reducible to 3D volume space, and "INSIDE" being perhaps even more mysterious... So we do not live in 3D. No. No. No... Have a nice day and take care

[LAChinthaka]

Thanks. Really good video & it helped me a lot. (Y)

[pickeyberry]

Video Content

00:00 Introduction
03:16 Question. What is V⁴? How to visualize it?
16:52 Linear transformation
37:11 Another approach to im(T)
43:21 Important observations
45:51 Proposition