Introducing the Inner Product - A Fundamental Concept in Linear Algebra

[@MathTheBeautiful]

Go to LEM.MA/LA for videos, exercises, and to ask us questions directly.

[@wenweilin4794]

The best linear algebra teacher on earth is back!!!!

[@stearin1978]

wenwei lin Don't discard the strength of Gilbert Strang!

[@MathTheBeautiful]

On the contrary, Gilbert Strang is my hero (and teacher) and mention to this whenever I have a chance!

[@scitwi9164]

Gilbert Strang might be good at linear algebra, but he's not very good in teaching. You are much better in that. No stuttering, no chaotic jumping over different topics and different parts of the blackboard, everything is clear and in proper order :) All with good, visual, geometrical intuitions and clear explanations.

[@TheRealJavahead]

@@MathTheBeautiful "But Grasshopper, someone must snatch the pebble," said Gilbert to Pavel. Agreed, Gilbert Strang is a legend. His OCW lectures were my introduction to linear algebra.

[@grantkobe9]

@@MathTheBeautiful Gilbert Strange is my No. 1 hero in algebra also . You are my No. 2 hero now ! Thanks for your teaching. Learn a lot from you.many thanks.:p

[@theodoretourneux5662]

this lecture is more engaging than anything I've seen before, it really does make everything sound beautiful! Thank you for brightening my day and bringing a smile to my face!

[@MathTheBeautiful]

Thank you - deeply appreciated!

[@raizan1526]

you're the best

[@MathTheBeautiful]

Thank you, it means a lot!

[@gianlucacastro5281]

Only by this intro I can be SURE this is going to be one of the best linear algebra material on youtube.

[@MathTheBeautiful]

Thank you, it's very nice of you to say!

[@miikavuorio9190]

May I reccomend, 3blue1brown

[@farrukhsaif108]

@@miikavuorio9190 They don't cover inner products

[@marciofernandes7091]

This teacher is something else. Thanks for posting this.

[@damian.gamlath]

This is what is missing from most textbooks and even YT videos - the reason why - the intuition behind the math and calculations.

[@anjanavabiswas8835]

Ok this is like the best lecture. He actually motivates his explanations. Even me with my 2 braincells can figure out what he means. When he gives the length of the polynomial example, it really helped me to understand why I can't directly measure length. The intuition was very valuable. Thank you.

[@alexplastow9496]

This guy lectures with all the conviction and zeal of a campaign speech, except it's math, which is fun and not ideologically poluted

[@MathTheBeautiful]

^This guy makes really accurate comments

[@jvmguy]

I really like the way you teach this. I thought I knew linear algebra, but this takes things to another level.

[@MathTheBeautiful]

There's always another level!

[@hedgeclipper418]

came here for a quick review of inner products and got this. I think I am happy with this outcome.

[@andreypopov6166]

After reading my current textbook and didn't get a lot, was surfing youtube for an explanation why Inner product is needed and it seems that this is the vide i was looking for. I believe the worth trying resource for sure. Thanks!

[@MathTheBeautiful]

So glad you found it helpful!

[@sanjinred]

Trully the best way to approach linear algebra of vector spaces. Not to teach how to solve it, but to actually give a deeper understanding of WHY we are doing it. I am a structural engineer and had to learn it the hard way, on my own because in college we only learned how to do it. :) Great vid!

[@MathTheBeautiful]

Thanks - much appreciated!

[@AyoubChouak]

Absolutely brilliant, so brilliant I went so far as to buy your book "Hello Again, Linear Algebra". Thanks for these wonderful videos and I wish you all the best for Lemma.

[@wuzark]

WOW, this lecture is really good. Thank you.

[@TheGodSaw]

Yes TWO new series. I love your videos!

[@irtizahasan3537]

it's so nice of you made these videos. thanks

[@andrerossa8553]

Thank you so much for such a great enthusiasm to teach

[@boybawn67]

Please consider doing a video on weighted least squares to show how the projection is oblique under the standard inner product, but orthogonal under the 'right' inner product.

[@evertonsantosdeandradejuni3787]

that'd be very intresting

[@jgarbs6468]

David Wallace is a pretty great teacher!

[@MathTheBeautiful]

and accountant

[@tcreatesllc]

A very good orator. Perfect

[@AlphaHatsuseno]

Holy moly this is amazing quality

[@sollinw]

not only did I understand what I didnt understand, but also understood it thx

[@زينالعابدينماجد-خ1خ]

I love how you teaching thanks for this amazing videos

[@anuragkadam7935]

This video was awesome, its like watching a suspense movie

[@MathTheBeautiful]

That's how I see it too!

[@rohamjarah7092]

LinAlg day 1: Solve for x and y. LinAlg day 30: How long is turquoise?

[@earlofyarg]

incredible teaching.

[@komahanb]

You are right, we have always been trained to assume "inner product" as just "length". Inner products, as you mention are far more fundamental than attributes such as lengths, angles (for geometric vectors). The nature, perhaps, must be using inner products to compare two objects (A, B) with respect to a chosen set of attributes. In the case of geometric vectors, objects A and B are vectors, and an attribute that we "chose" to do the comparison is length. If we compare two surfaces A and B, the attribute perhaps can be chosen as area. If the surfaces are identical but differ only in roughness, then choosing just area wouldn't suffice to tell whether A and B are identical or not. Then we have to compare both area and roughness. If two surfaces A and B have the same area and also roughness, but differ only in color, then we need to include color as an attribute for comparison.

[@MathTheBeautiful]

Hi Komahan, thank you for a very interesting comment. However, I'm quite confident that nature doesn't think about inner products. -Pavel

[@voidisyinyangvoidisyinyang885]

check out Alain Connes on noncommutative spectral that is nonlocal inner products. thanks

[@jaimelima2420]

Where are just watching the birth of the inner product. And his mom is called Norm. Thanks for putting this together.

[@nielsota63]

Hi! I love this video series! I was wondering if you have any exercises to go with the videos?

[@MathTheBeautiful]

Yes. lem.ma/LA

[@umbraemilitos]

Inner product spaces are just a special case of tensor spaces.

[@shivanisingla1140]

You are an incredible teacher🤗

[@defaultuser1760]

Amazing explanation. Thank you.

[@worldmath8848]

thank you so much sir for adding such a nice video ... Keep it up ....

[@kushalv8268]

Thank you sir for amazing lecture

[@Hythloday71]

Is this the beginning of a new higher course in Linear Algebra ? Oh goody !

[@omkark7597]

Prof, Video is great. please publish videos on dual spaces.

[@Matchless_gift]

This 14min lecture can clear purpose of doing l.a

[@avtaras]

Looking forward to this :)

[@MathTheBeautiful]

Do it on Lemma! lem.ma/LA (and lem.ma/LA3 to jump to inner products).

[@TheGodSaw]

Hey, you said you would talk more about the SVD and its application. Will the be in the context of Inner products?

[@MathTheBeautiful]

Yes

[@alexshei5061]

Thank you so much for these amazing videos!!!!

[@MaxPicAxe]

Very good video

[@solarestone]

Thank you

[@MathTheBeautiful]

Glad you enjoyed it!

[@levtunik997]

great editing!

[@AliVeli-gr4fb]

i am excited

[@mrcaljoe1]

brilliant video. wish he was my lecturer

[@MathTheBeautiful]

I *am* your teacher. Just check out lem.ma/LA

[@ratusca]

@@MathTheBeautiful Thank you so much, my concepts are so clear after watching this video. Online classes are useless :(

[@lateefahmadwanilaw8948]

Thank you sir

[@doodelay]

Damn this guy is good

[@MathTheBeautiful]

Correct

[@Euquila]

2:18 that lambda is more like a giraffe no?

[@MathTheBeautiful]

Yes, yes it is

[@shawheennaderi8970]

For some reason, his teaching style reminds me of Richard Feynman's

[@VNischal]

I Totally Agree..... :)

[@jsnam8139]

Might be because of his accent.

[@alexbenjamin5823]

Would factorizations fall under II? (eigenvalue, LU etc.)

[@MathTheBeautiful]

Depends on the factorization! I: LU, LDU II: XΛX⁻¹ (Eigenvalue) III: QR, LDLᵀ, LLᵀ, Polar, XΛXᵀ (Eigenvalue for symmetric), UΣVᵀ (SVD)

[@RahulYadav-nb2zt]

very nice lecture

[@RahulYadav-nb2zt]

very nice lecture

[@_computerra]

I wish I was in your class.

[@MathTheBeautiful]

Thank you for the compliment! Check out lem.ma/LA and you'll feel like you in my class.

[@phenax1144]

love it😀😀😀

[@ashwinsingh1325]

Solid lecture

[@adrianott5248]

What makes it so obvious that length is the right measure of how accurate the (semi) solution is in the case of your rectangular matrix multiplication? Why not minimize the sum of the errors? I know its a more convenient calculation and it uses the power of matrices, but is that the only reason?

[@MathTheBeautiful]

You're exactly right. There isn't one best preset measure to be minimized. The choice of measure should depend on the particular problem you're trying to solve. Whatever measure you choose would be called "length". Some lengths come from inner products, some (like the sum of |errors|) don't. The ones that come form inner products have some advantages. Other measures, like the one you're suggesting, have other advantages.

[@adrianott5248]

Thanks you! Looking forward to more videos from this course!

[@spearius9059]

I really want to be in your class.

[@MathTheBeautiful]

Please!

[@stanleyezeogu9816]

Wow!

[@MathTheBeautiful]

Glad you liked it!

[@JohnBerry-q1h]

. . . . . . . ** . . . . . . . . ** What is☝☝☝THIS or ☝☝☝ THIS?? I often see this notation in mathematical writings. To me, they both look like inner products, but with THREE inputs. How do you go about evaluating these? What is the proper interpretation of this notation?

[@MathTheBeautiful]

See kzbin.info/www/bejne/hqTFYo2rrdRqndUsi=3E1stR16in-S0gyD&t=250 for one possible analogy. Another analogy is that it's the inner product of the vectors 𝜓 and (𝚽𝜙)

[@johnnymurf]

Innah prawduct

[@achillesarmstrong9639]

good video

[@anuragkadam7935]

You sound so much like Anthony Jeselnik!!!

[@MathTheBeautiful]

I'm going for Mitch Hedberg actually.

[@111abdurrahman]

Great actor. You need to join hollywood

[@MathTheBeautiful]

Thank you! Please tell me you pronounce your last name "Riemann"

[@voidisyinyangvoidisyinyang885]

"orthogonal projection" is two words - just sayin' haha

[@goobersteinmcfancyful]

Everyone is shy, so I'll just say it... much better than Gilbert Strang.

[@MathTheBeautiful]

I'm not sure I agree, but I certainly appreciate the complement!

[@봉디옐]

Algebraists agitation