Transpose Definition

Definition of the transpose

Have you ever wondered where the transpose comes from? In this video, I show that the transpose arises naturally in the setting of dual spaces. This should also illustrate why dual spaces are so important. Enjoy!

[ddiq]

“Now what the hell does that have to do with matrices?”

You took the words right out of my mouth.

[maurorafaelfelixcastaneda]

I'm studying to enter a masters in applied math and your light-hearted videos in topology, analysis and linear algebra are helping me a lot! Sometimes I get overwhelmed with the tons of information and whatching you teaching these subjects with a smile in your face just keeps me on track. Thank you Dr. Peyam!

[XanderGouws]

I like to think about the transpose as turning all of our rows into columns and columns into rows.

[iamreallybadatphysics]

Dr Peyam I adore your videos so much. Everytime I look at a new one, I want to learn more math, it is really beautiful thank you!

[LarryRiedel]

Transposes and dual spaces are used all the time in the real world, so any intuition helps. Given a vector (x0, x1, x2, x3), what is an equation for the hyperplane it is perpendicular to? What is the relationship between the scalar x^Tx and the rank-1 matrix xx^T? Understanding these things pays dividends.

[him]

Dr Peyam, regarding the comment on this video about hyper planes - extremely briefly, and I say this as someone who hasn’t finished your dual space playlist, but who is very grateful that it exists!! (Never delete your videos) I wonder how hyper planes to a vector have anything to do with abstract functionals? A sketch of the link would be greatly appreciated. Love your work, channels like these have taught me all the maths I know pretty much

[ziminfan]

Keep the great work, please! I think what u taught is 2nd course in Linear Algebra, which is very proof-based with little intuition in many Universities' math department. So, these video is really unique and helped me a lot in terms understanding the proof based math.

[navyawhig]

THANKYOU SO MUCH SIR EVERYTHING JUST SEEMS SO CRYSTAL NOW

[technofanatic]

Dr. Peyam, you are my hero. thank you so much for your wisdom

[EnsnaredProductions]

Hi Dr. Peyam,
could you make some videos about tensor products and the vector space ofalternating multilinearforms (generalized determinants)?
Lovely videos!

[waterfirecards]

Thank you very much, I really appreciate what you do !!! Thank you!!!

[cezarionescu]

Damn, that was a nice and relatively rigorous proof, you sold me on your content, will watch it all

[davidgould]

Sorry for spamming, but I just wanted to say that I a) really liked that; b) will have to watch again (at least once) because I i) really liked that; ii) am really not up to speed with some of that, but I'm getting there.


Thanks for the great videos! Keep them coming (please).

[lynny]

15:55 Why did you multiply ∑(from k =1 to m) A_ki with w_k?

[localfuture]

Dr Peyam, I am a little confused for the T^T(gj)(vi) = gj(T(vi) at from 14:33. Here we want to prove T^T(g)(x) = g(T(x)). Is that a little kind of "circular reasoning"?

[mmmgbmmm]

If the field is over the complex numbers, does that mean we must replace 'T' with 'transpose conjugate'?

[BALAGE]

Is the matrix of a linear transformation a "row of columns" or a "column of rows"?

[murilopedroso]

Heyy! Thanks for the video! I'd like to know if is there any book which you recommend for matrix and linear algebra, because all i know don't show this amazing perspective?!

[sajidhaniff]

That was awesome!
How do you correctly transform a normal vector to a plane as it a row vector which is a member of the dual space

[him]

Also, you say that we want to show that the transpose matrix is also a transpose in the dual space definition, but the way you phrased made me wonder if there exist other ways to find a “transpose” of some matrix that isn’t actually its transpose matrix!