Tensors for Beginners 8: Linear Map Transformation Rules

ERROR: At the end of the video, the Linear Map transformation rule uses indexes "s" and "t", when it should use "i" and "j".

Going to try to upload the next video on the weekend.

[eigenchris]

A lot of people are confused about the FLB vs BLF formulas at 3:12.

The order of matrix multiplication does matter. If F, L, and B are matrices, then FLB is NOT the same thing as BLF. But I have not written matrices on this slide. I've written F(^j_i) L(^k_j) B(^l_k) = B(^l_k) L(^k_j) F(^j_i), which are matrix components (ordinary numbers). You can switch their order just like you can re-write 3*4*5 as 5*4*3. The matrix multiplication happens in the summations over j and k, not the order of the terms in the sum. It's the summation indices themselves that you are not allowed to change.

[shiroshiro]

I’m an idiot (being part of the undergraduate system)to study general relativity on my own and your videos have save my life and fulfill my mathematical(and physical)intuition. Thank you for the very good work.

[jamescook]

Nice work. Having lectured this stuff once or twice, I am impressed. It's a pain to get everything said without accidentally changing notation midstream.

[uhbayhue]

I was going to wait until the end of the video series to comment, but I want to say that I'm so grateful for your explanations. They have great intuition and well mathematical rigor as we prove everything we use. And the examples are amazing. All in all, your presentation of this material is so enlightening, especially for someone trying to self-study. Thanks so much!!!

[TheTck]

Chris, thank you for the time and effort you put on your videos! Our professor used one slide to explain tensors and then moved on to more advanced stuff with them, so needless to say no one in the class had any idea what was going on. I have learned so much more from this series than I have during the last month in differential geometry. Thank you!

[okcmqkd]

This series is great. Thank you so much. I'm a physics student, and this comes up a lot, but was never explained properly. I will tell my friends about this series.

[cermet]

Again, outstanding and rather straight forward on a subject that never appears so when covered in std text books. These are critical concepts rarely covered much less in such a concrete manner; your basis vector example was very insightful and really helps explain what was going on in the math.

[fallon]

super clear and I regret not self learning in the summer. thank you very much. you are a great teacher. ❤❤❤❤❤❤

[JingxiangZou]

I came across this video series this Friday when I wanted to learn about tensors and I have been spending the weekend on this series. I found it immensely helpful and very well organized. I just wish I had known this video when I was an undergrad. I am eternally grateful for the hard work the lecturer has put in making these videos. I hope one day when I become an expert in a certain field I shall too make videos as high-quality as the works by this lecturer to help people around the world. This is one of the coolest things one can ever do! Thank you so so

[adriander]

In case anybody is as confused about the ordering of the bases and components at 11:04 it might help to realize that the bases and components in the same row don't belong to each other. Rather the table lists the vectors that behave in a covariant or contravariant way with regards to a change in basis vectors.

E.g. in the first row we have the basis co-vectors on the left but the vector components of the normal (contravariant) basis on the right. In the second row, we have the basis vectors on the left but the co-vector components on the right.

Probably for most people that was already obvious but for me, it took a while to realize that. I hope it helps some of you tho.

[mranonymous]

I wouldn't call the Kronecker delta an "index canceller", but rather an "index replacer". For example at 8:42, I would say that instead of cancelling the j indices, the Kronecker delta replaces the repeated index j by its own non-repeated index k, so that you end up with M^i_k instead of the original M^i_j

[HualinZhan]

A suggestion: if we consider L(v) as (invariant) vector in both old and new coordinate systems, we can quickly obtain \tilde{L}=BLF. Specifically, \tilde{L}(\tilde{v}) is simply L(v), an invariant vector with components that are now described using new basis, i.e., \tilde{L}(\tilde{v})=BL(v). Put v=F(\tilde(v)) in this expression, we have \tilde{L}(\tilde{v})=BLF(\tilde{v}), i.e., \tilde{L}=BLF.

[AntKPro]

OMG! After all the years wasted, I start to understand why we need matrix decomposition and other linear algebra tricks!

[amitmodak]

best explanation of tensors and all gr prerequisite anywhere i can find

[TheBigBangggggg]

Wonderfully explained! I surely hope you will continue this series ... one day ;-)

[cwaddle]

This reminds me of the first year linear algebra class that i so struggled with. The symbols of the products of backward and forward operator ... but i think i follow it a bit better

[abnereliberganzahernandez]

nice Videos. being said that i am really mad at people writing about this subject here in this series of videos you explained all about covector and linear maps in a very formal way but very simply explained so we all understand and I had issues about the einstein convention becouse some books just said that the sumation sign goes away when having dummy indices an contrary to free indices, but explained that way this convention seemed ridiculus beocuse they did not explain its procedence.

and I really spended weeks reading books. but finding your videos makes me happy but at the same time mad at writers who cant write good books on the topic.

[JgM-iejy]

So far excellent work explaining clearly what a tensor is, as opposed to horrible math books that only offer incomprehensible explanations. This from someone who had always used vectors as mere pointy sticks.

One thing missing so far is the WHY of it all. WHY would anyone care about changing basis ? What is the whole point ?

[ghuhkuh]

Hello Сhris, First of all, I would like to say thank you for your incomparable work. I would like to know if only I have no subtitles in this video or is it disabled on your part? (But there are subtitles until Lesson-8).

[sajidhaniff]

Nice explanation of similarity transformation. Thanks