Tensors for Beginners 12: Bilinear Forms are Covector-Covector pairs

At 7:04 there are mistakes in the 3rd and 4th lines... Some of the B indexes are reversed.

So I got a new mic. It probably sounds like I'm 5 different people in this video because of the re-recordings I did. I hope it's not too distracting.

[eigenchris]

Around 6:45-7:04 there are mistakes in the 3rd and 4th lines... Some of the B indexes are reversed.

[Charlesstrahan]

I almost never comment on videos, but I just have to say: this series is awesome. Thanks a ton for putting your time and energy into producing these!

[Hermis]

As an aircraft engineer, I was only used to the length-preserving transformation of three-dimensional vectors. But I recently started to think that, the tensors, which I had an only taste of long ago, maybe a good tool for solving control problems of complex nonlinear systems. I decide to learn what it is and these videos are amazing to me in many aspects. Each stuff explained is so easy to understand. Thank you!

[jamesbaxter]

This series is the best expanation of tensors and their notation, you make everything very clear.

[TheAntonlife]

I was so confused about how to formulate the metric tensor as being sorta double covectorish and now I finally understand! It was my reason for starting this series but now I'm gonna watch the whole thing

[Nickelicious]

I’ve literally been searching for an explanation like this for an entire week and now I’ve finally found it! Thank you so much!

[erikengheim]

Amazing job you have done here. This must have taken a long time to create. You put a lot of attention into using colors in a helpful manner in showing the math. Also you have been very thorough in explaining all the concepts that those of us with with patchy memory of our school math can still follow the video series. Very thorough and carefully planned.

Good pacing, editing, clear voice and everything. Very well done videos.

[dhwaneelkapadia]

I feel like I had my first Eureka moment while watching this video. Thank you for the amazing content.

[haibojiang]

I tried to build a constitutive model for geotechinical material and promote it to general stress space, which requires a lot tensor knowledge. I have been looking for different books and videos. These videos really inform me about what is a tensor and how it transforms. Your video is highly appreciated!!!

[awakening]

I like the idea of "row of rows", which disclose the awkwardness I have never noticed before. Thanks.

[rlicinio]

I could not understand a single word for the last three or four last videos but strangely I am still enjoying. Thank you¡

[subrotodatta]

I am struggling with this derivation between lines 2 and 3 where the BijVk products are being matrix multiplied and regrouped as a row "vector", with two entries of row vectors. I am aware that there is a typo in the B12V2 term which should be B12V1 and the B21V2 term should be B22V2.

[Drull]

Thanks a lot for the video. It's literally a godsend for me, as I'm struggling with the tensor in general relativity course

[zeotex]

Bro... this feels like completing my basic Linear Algebra knowledge from undergrad math. How did we learn none of this :( <3 <3

[adarshchaturvedi]

Around 6.50, we have row of rows, that's a 1x2 matrix ryt, which when multiplied by a vector ( a 2x1 matrix ) will give a 1x1 scalar .. which we get when a covector acts on a vector .. after that, scalar multiplied by again a vector which is (w1, w2), how it's giving a scalar (

[danielribastandeitnik]

Really nice the last part on row of rows!

[isaacAdam]

Thank you sir I love your work I did understand tensor meaning from you for the first time

[mot]

Really enjoy your videos. Had the same question as below (at 6:45)..glad you said it was a typo--I thought I was very dense for about 30 minutes trying to figure out what I was missing. I feel often times I am dense but not that dense....

[likeplayinghello]

Around 6:40, I cannot understand when you start to multiply B and the column vector v. There is a + sign that disappears on the next line. Could you explain more ? Thanks a lot.

[steveparsley]

I have been trying to see how the bilinear form, written as a row of rows, can work with the two forward transforms written as square matrices, to yield the "new" bilinear form. I can never get the same result as I get when I use the formula at 4:25. Using the formula, for B11~, the F terms would both have '1' in their lower index, but I always get some terms with a '2' there. Am I just being clumsy?