Inverse matrices, column space and null space | Chapter 7, Essence of linear algebra

学了一个月才看到😢

When discussing rank, does something collapse to a single point have a rank of zero?

If only i could explain how much happier and relaxed i feel after understanding Ax = b intuitively. :)
This series is a gem! Hats off to everyone involved for making this accessible to all.

압도적 도움!

My comment will be another in a sea of many, but I wanted to express my gratitude for your existence. Your goal of giving maths a more intuitive approach really helps people like me who don't like to accept rules and formulas as they are and just use them. Please keep the good work up, it is invaluable.

I fucking love you

Hey congratulations. You are president

This series is pure gold. I can't get enough of it!

Please You may publish a video about the kernel of function (for an example
1/(2pi i) integration of e^(2pii)*f(x)dx

these videos need oscar🤯

Hi, not sure how you got the inverse values of a clockwise triansformation that sends i hat and j hat back to starting position. I tried negating all the values and got them. Not sure if this is correct? Looking at shear example I see just a negation of a x coordinate of j hat therefore move to the left whtch makes sense and y coordinate stays the same(1).

Grant, I hated maths since middle school since I did not see my teacher understand it or pass intuition along, I was left with a bitter taste of maths as my teacher put me in a lower maths class many years ago as I refuse to complete my homework for pointless learning. I almost did not make it to university because of his decision.

Luckily, I pulled through with the best grade even in my lower class. I took it for granted that maths is something for people who enjoy abstraction and detached from realities of society. It was only in university where our scientific lecturers explained to me why we needed to model, and why we needed to use mathematics as tools to make models, and starting from philosophy of why we do maths and science again was I able to established a loving relationship with maths. I have since completed an MSc. degree through teaching myself maths. Your videos have been immensely helpful for someone teaching themselves what people have learnt brute force in middle and high school. I caught up with a lot even compared to my university classmates who aced maths class the traditional way.

Just shows you how much you really should understand intuitively before teaching someone, I felt like a lot of our education system around the world has loss their way in comparison. We should be educating young minds up from curiosity, philosophy of knowledge and science, then proofing from intuition. Your videos have been fundamental to my scientific journey and will continue to be in the future. Many thanks for your thankless work. You are legendary. I am so grateful for all you have done.

I’m never going to lecture again

Can you make a video pls about hessian vector products? I cant wrap my head around it 😢❤️

wait this isn't sonic forces

Your work brings art to mathematics in a such beautiful way. =)

These series of videos are brilliant. From the moment each video starts till the end it invites all to think and understand. You're great.

This is pure gold.

I'm not lucky enough to watch these videos before going through linear algebra at university, now, I'm watching these and gradually understanding what I had learned in the first year, but I'm still feel glad. Thank you so much.

for pre-colleges,it's still too abstract,even though it's much easier to comprehend compared to other series😢

I am reborn after watching this playlist.

So the matrix multiplication is not comutative exept for its invers? Is that right?

공부하고 다시 보니깐 또 새롭네

I feel something's wrong with me or something? to everyone else this is more intuitive than uni but this just confuses me more.

Should I watch essence of linear algebra series after or before learning linear algebra?

Doing this course with 18.06 is the best decision I took in a while.

You are my hero. You fundamentally changed my relationship to math. I was reading some papers trying to understand this topic for some hardware tinkering, and I was just hoping there would be a video by you!

why cant you unsquish the line when you clearly can

Thank You!

Gonna do crack about it

I was struggling so hard to understand what the null space was until I watched this. This is awesome

i didnt get what is the difference between "span" and "column space".

so thats why a matrix whose determinant is 0 is called "singular"...

If I had known this before going into the university, I might have actually learned something.

This series has really 'transformed' my understanding of linear algebra.

This is pinpoint to the things that are lacking in most education systems. This type of thinking is the future and we will look back on how we used to teach and it will seem old and flawed.

dude where were you when the westfold fell

Please create a video for visualizing the calculation of inverse of the matrix, and what do minors and cofactors mean visually. (3x3 matrix requested)

I think I have a suggestion on how to get a 4D depth, of an oblique cube.

I've always wondered if there is, another way to get 3D depth, than just changing the angle of the z axis.

After many failed attempts, I have come to the conclusion that if you change the angle of the z axis,
and at the same time stretched the cube on its width (along the x axis) of the left stereo image, so you could even then get depth!

At first, I was very doubtful, as to whether I had really achieved 4D depth, because the cube looked like it had normal 3D depth, and no extra depth at all, since I only changed the x axis (width).

So I tried to make a flat field, with a cube on Paint by copying the cube. Then I placed a cube on the corner, in the front row on the left.
Then I placed another one, on the corner in the back row, on the right. To my great surprise, it turned out that the distance between the two cubes actually had depth!!!

Below I have made a stereogram (3d image), of line symbols, to show how it looks :


| | | |

By drawing in the lines along the width, in the left picture, you can get extra depth.

This made me realize that if you wanted to get the w axis, you had to use the x axis, along the width. It is not possible to exclude the x axis!

We may not be able to see 4D entirely, but we may be able to see 4D depth!

I would be very happy, if someone wanted to make a isometric game, like that!
It would also have been cool if you saw it in perspective!

Thank you for reading this!

:-)

This series is something I was looking for last two decades.

I came here while searching for role of Eigenvalue in dimension reduction.

I think most of the people who have studied matrix will agree, that we only know the calculation part but what's the geometric meaning, nobody knows.

Me at my orals: no I said you do it with a computer, I swear.

THANK YOU