Orthogonality and Orthonormality

We know that the word orthogonal is kind of like the word perpendicular. It implies that two vectors have an angle of ninety degrees or half pi radians between them. But this term means much more than this, as we can have orthogonal matrices, or entire subspaces that are orthogonal to one another. Beyond this, we must understand the term orthonormal, and why orthonormal sets of vectors are important. Let's check it out!
Script by Howard Whittle
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[@pkasb90]

I must say your lecture supercedes those in higher institutions.

[@STKeTcH]

same

[@miguelnuno928]

Correct

[@BoZhaoengineering]

agreed.

[@overl0ad816]

🤓

[@julenekenyon3278]

I agree 💯. At this point you deserve my tuition!

[@isxp]

Good god, I wish I found this 8 weeks ago. The drop date for classes is the 30th. I ended up with 6% on my first midterm in Mathematical Physics. This class will haunt till the day I die. I'll probably know this material better than any of the classes I've taken, as I'll likely obsess over it for months.

[@kartikkalia01]

I'm at same situation as yours, please suggest similar videos/playlist.

[@mohammedbastaki2101]

same here pal

[@lingwaili1203]

Thank you so much! Professor Dave explains clearly so I can finally understand orthogonal

[@ddiverr]

clean, informative, and concise video, thanks guy

[@missghani8646]

can't thank you enough for this clear explaination

[@tadabae]

my final exam is in 15 minutes and i stumbled across this channel. he explains this so clearly!! i wish i found this channel earlier omg

[@beams9497]

Haha saaaame

[@macdonaldnnadi]

15 minutes is crazy dawg

[@bluefenix1457]

This explained it so well for me, you spoke clearly and didn't do messy sentences, and even paused after every sentence to process it 👏👏👏👏👏👏

[@janakamohotti]

Brilliant explanation!!! ❤️

[@user-qv8mr8pe3i]

thanks for awesome explanations!!!

[@jainpravachan650]

Thanks sir.... wonderful lecture

[@jisaburo198]

EXCELLENT videos! Thank you so much

[@ayazohdy6221]

Professor Dave explains 😂💕

[@rodab3546]

you explain so good

[@manishbhatia2724]

Beautiful Explanation

[@JoseLopez-op7sq]

A lot of good information in one short video; good overview.

[@Dixxi91]

Very good explanation, thank you

[@raihansk5971]

Amazing!!.........Explaination is awesome.....

[@mehuladhrit7883]

Very nicely taught...

[@Bruhhhhhhhhhhhhhhhhhhhhhh]

3:05 "four SQUARED plus two SQUARED plus negative one squared" lol im dying and good vid overall

[@Robert-gr1cl]

such good explanations, thanks

[@apythonprogrammer7877]

This really helped me understand LLM model quantization just a tiny bit better

[@blakeelzinga1168]

Thank you for this video

[@tknam3278]

This is perfect!

[@adefisanadedotun797]

Very understood
Thank you

[@user-so7gh7gd4v]

Amazing explanation

[@suvarnamanojsingh]

thanks that was helpful

[@mona5112]

Hello Professor and Thanks for your great explanations.
I was wondering why do not we have something called orthonormal matrices ??
and think orthogonal matrices are more like orthonormal ones!! :))

[@lamdawave]

Good video. One question: If a square matrix has orthogonal column vectors. its inverse is not equal to its transpose. what should we call this type of matrices?

[@mcalkis5771]

Could someone elaborate on the weight functions? Is it just a correction factor so that a function can be orthogonal with respect to another?

[@jds189]

what happened to the visuals clarifications? its been primarily plug and chug for most of linear algebra..

[@umehmoses8118]

I love the professor 😁

[@colingillespie7635]

The intro alone earns my like

[@banderallogmany1912]

3:52 I didn't get how the length becomes 1!

[@deathworld5253]

Isn't orthagonality defined by having the dot product equal to null element in Euclidian space?

[@yogsothoth2194]

I think null just means nothing; or in mathematical terms 0 . So yeah you're probably right

[@asifnahyankabir7747]

Amazing Professor! One day if God wills I'll come to meet you!

[@kidatheart4975]

I'm confused on the orthonormal part. There are 2 conditions for orthonormal vectors: (1) orthogonal; and (2) the length is 1. But the example on 2:56, the length is not 1 that negate the conditions of being an orthonormal. Can you please elaborate that part? Thanks

[@criclal1787]

Those vectors displayed at 2:56 are not orthonormal, we have converted them both to unit vectors and made them orthonormal thus.

[@wealthy_concept1313]

@@criclal1787 but if that's the case it means any vector can be converted to orthonormal at will ?

[@MuffinsAPlenty]

@@wealthy_concept1313 Any set of vectors can be "normalized" (meaning to make the lengths of all of the vectors 1). This does not, at all, change the angles between any of the vectors.
The Gram-Schmidt Process (the next video in the playlist) shows us that any _linearly independent_ set of vectors can be made orthogonal without changing the span of the set.
Taken together, given any basis, we can always find an orthonormal basis by first using the Gram-Schmidt process to make the basis orthogonal without changing its span, and then we can "normalize" the orthogonal set to make it orthonormal.

[@ManojKumar-cj7oj]

Inverse of orthogonal matrix = tranpose of matrix

[@dgormley]

Please teach at my university.

[@kryptoid2568]

sir, the inner product notation reminds me of bra-ket notation

[@kaanylmaz1008]

eyw reis

[@prakashraj4519]

video on caley hamilton

[@srinityapadma5125]

Thank you so much Sir.../\

[@desrepeerc206]

3:23 Nice frankenbiting skills xD

[@glennveniceaboy8876]

your convention for magnitude of a vector is a bit confusing because the single bar on both sides is usually for absolute value, maybe you should've used double bars for it
anyways, i learned a lot, thanks!

[@carultch]

Absolute value and magnitude of a vector have so much in common, they might as well use the same notation. I thought the double bars on both sides was completely unnecessary, when I was first introduced to the notation, after having become accustomed to just using the single pair of bars.

[@anandailyasa2530]

❤️❤️❤️❤️❤️

[@Gafa996Gaddisa]

Is Ortho Greek word which means , straight up?

[@georgesimos4914]

"ορθό-ς" is used for other cases too; the one you say is one definition, but the one required for the concept of the video is "vertical"(an example is the mathematical expression "ορθή γωνία"="right angle")

[@carultch]

@@georgesimos4914 Even though I know that 3 out of 4 of the letters have completely different pronunciations, I instinctively read "ορθή" as "open". Even though I know it would sound more like "orthi".

[@rushdisami5146]

Finally I understood Orthogonality.🤖

[@user-iv4iy5rc1y]

TE DUA

[@marcosramirez2278]

I found this channel because of flat earth videos, never did I guess this man would save my math grade

[@jaliyahkane5127]

professor dave is an AI

[@aniketgupta5750]

Elaborate to some extent, I mean your beginning and laying down the foundation of the topic is good but should stretch it till good level.
Atleast that's what I feel missing in your videos, do please consider this if you see this comment.
By the way I love your videos from quantum numbers to biomolecules all are awesome.

[@rayandharia1718]

CHEMISTRY JESUS CUT HIS HAIR

[@garyallen538]

Yeah but....what does this have to do with birds? (Sorry, couldnt resist!)