Відео

Thank you for explaining concepts intuitively. I don't know why my professor cannot do that.

at 28:14 why are v1 and v3 free variables and not v2 and v3? Row 2 and 3 are all zeros.

Hi professor Johnston. I love your videos. They 've helped me a lot. Just a question : Why can we split matrices into smaller pieces - matrices and then do the multiplication? I mean the result is valid but i don't quite understand the process because these smaller matrices are not actually matrices, since they have no matrix brackets around them.

At 15:45, does the third matrix of the Pauli basis have a norm of root-2? It seems to me that it has a norm of 0, since we are adding together the squares of "i" (which is -1) and "-i" (which is 1).

Can you do a playlist on semidefinite programming or sum of square optimization things ?

At 8:40 you use this square-root transpose matrix to operate upon matrices from the STANDARD basis (rather than the Pauli basis). How do you do that? Do you have to use a change-of-basis matrix or something? And what is the matrix [a b, c d] in terms of this standard basis? EDIT: Ah, I think I see the answer. In the book on p49, you find the standard matrix of the square-root transpose, but built up from the matrices represented in the standard basis. So, this then operates upon matrices represented in that standard basis.

It would be nice to know more about what the implication of this is? Does it mean, for example, that there is NOTHING that you can do with, say, degree-3 polynomials that cannot be done with vectors in R^4? Seems like that surely is not true.

Thank you 😩

I'm trying to recreate this but the links are dead. Where can I find the pattern file?

I don't know how this man ISN'T FAMOUS!. great job Mr.Nathaniel. keep it going!!!

Thank you sir so much for your explanation

Why if <Tv,v> = 0 for all v, over field C then T=0?

Why have you stopped? This recent series of videos was interesting.

Thank you so much for all the lectures. True gems. I noticed you missed the negative sign on the 1,1-position, when re-writing the matrix E. Thanks again.

I'm a reader of your book "Introduction to Linear and Matrix Algebra" published by Springer. I love your book, which is amongst the best-in-class for someone like me who like to have more in-depth knowledge on linear algebra beyond those studied in high school. In particular, I love the section 1.4.2 "A Catalog of Linear Transformation". However, during my re-visit on this topic, I try to derive some of the formulae by myself, and am successful doing so except rotation in 3D. I'm able to get the exact formulae for Ryz and Rxy, but not Rzx which I have the signs for those "sine's" swapped. I observe you've put some remark at the margin to use Rzx instead of Rxz, but not sure if this is correct. My approach is: if Rzx means to have y being the axis of rotation, and positive z rotates towards positive x, then Rzx(e3) would become (sin(theta),0,cos(theta) and Rzx(e1) become (cos(theta),0,-sin(theta)). More or less, by analogy to the 2D case (by ignoring y), z replacing x and x replacing y. As a result, I have the signs of those sine's different from yours. Would you please shed me some light on where my approach goes wrong. Thanks a lot!

Thank you very much for a clear explanation!

Hi I'm trying to do the reverse echelon form for the 5x5, but I'm unable to replicate your chart. My efforts gets me to row 21 which has no zero and no 1's in the subsequent rows in the same column, so I'm unable to flip it. I'm able to do it with a 3x3 though. Is there a walk-through of how to do the row echelon for the 5x5?

Hi, I know this was a while ago, but do you plan on doing any lectures on multilinear/tensors?

Everything is explained in such an intuitive down to earth manner abstract things become straightforward very quickly. This is excellent.

Idk if you still active, but this helped me a lot in Linear Algebra 2 course, do you also have videos about orthogonal diagnolization etc?

6:51 breakfast is ready

Thank you!!!

Thank you!

I don't know why I'm watching this, but it was entertaining as hell.

Save my life

Excellent!!! very very good!

Let me just first say, you are a GODSEND! I cannot believe how well you explain the concepts and the way you relate everything geometrically truly aids in the learning and understanding process. Again, I cannot thank you enough for what you do. I am teaching myself the entire undergraduate curriculum for mathematics and just got done teaching myself Calculus 1-3 and am now tackling linear algebra before moving on to differential equations. I finally feel as though I am making progress, much more so than when I started studying the subject using Howard Anton's textbook. THANK YOU THANK YOU THANK YOU!!! You have truly been blessed with a gift for teaching what can be a complex subject matter in a truly understandable and engaging way that facilitates the learning process and makes for a deep and intuitive understanding of the subject matter.

There are unknown way to visualize subspace, or vector spaces. You can stretching the width of the x axis, for example, in the right line of a 3d stereo image, and also get depth, as shown below. L R |____| |______| TIP: To get the 3d depth, close one eye and focus on either left or right line, and then open it. This because the z axis uses x to get depth. Which means that you can get double depth to the image.... 4d depth??? :O p.s You're good teacher!

In every single other math book that I have found the inner product for complex numbers is linear in the first entry and the second vector is conjugated. This is the case for the Wikipedia entry for inner product, in Luenberger's book on Optimization, in Sheldon Axler's book "Linear Algebra Done Right" and in Byth and Robertson's book "Further Linear Algebra". Is this a quantum mechanics quirk? I cannot get the standard Gram-Schmidt formula to work on complex numbers when I use your method.

Hi Sir thanks for these exquisite lectures. I have a question regarding null space. If we are given set of vectors from space of 2x2 Matrices how can we find null space of that, thabks.

Hi Sir , it is great to watch your exceptional lectures. I have a question that how can we find null space of vector space of 2x2 matrices. Please explain thanks

He looks like Alexander Mcqueen!! Love the lectures

your videos are addictive :P

If B is a subset of V then span(B) is a vector subspace. then what is the span of the empty set?

Dude been fascinated by conways game of life - found ur vid. Great vid. Thought i was the only one so fascinated

your face blocks the bottom of the page 7:50

Great analysis of linear combinations. At 3 minutes, equivalently , we have to show that (1,2,3) is an element of span( { (1,1,1) , (-1, 0, 1 ) } ) , thinking of "span( ) " as like an operator (or a function), e.g. you give the span( ) a set of vectors as its argument, and it spits out all the different possible linear combinations from those vectors. (Again, at the risk of sounding repetitive, we have to show that the vector (1,2,3) is an element of the span of vectors (1,1,1) and (-1,0,1), where the span is a 2-dimensional subspace of R^3 , i.e. a plane ).

please fix the controls. the webside is giving me a headache but i love conways game of life

Why are you so awesome! A lot of work these videos and you hardly get any views. Just started your Conway book and having so much fun!

Thank you so much. I'm working through a scientific computing textbook right now, and they do not go easy on you with the linear algebra. Very intuitive explanation.

Hey king, have you thought about covering Measure Theory in a similar manner to how you covered Advanced Linear Algebra 🫣

I think that this was a great explanation for quadratic equations, but it’s not generalized if you don’t know the order of the polynomial, right? We can’t make the assumption that we have enough points to solve unless it’s explicitly stated in the question (in which cases we have at least enough and reducing solves it)- but we normally can’t in the real world.

Brilliant explanation! I have only one complaint. I have read (though, not thoroughly understood) both of your linear algebra books and your point in the beginning of this video of "flipping your thinking", pointing out that x and y are constants in "linear world" would have been quite helpful in my early attempts at understanding linear algebra. My last formal math class was about 50 years ago so x,y and a,b as variables and constants are pretty much hard coded in my brain, so I need all the help I can get. In all seriousness, great books and great videos!

Once again Nathan, you have outdone yourself with video quality, this is amazing

Nice! (I’m hotcrystal0)

Nice video. Was informative and you have good cadence

Thank u so much!

Thank You !

I paused the video. My guess is to make one circle as big as possible (touching all sides of the triangle) and fit two more in two of the "corners" touching the big circle and two of the sides of the triangle. Now I resume the video to see if I'm right...

Please cover abstract algebra sir 😭