18. Properties of Determinants

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MIT 18.06 Linear Algebra, Spring 2005
Instructor: Gilbert Strang
View the complete course: ocw.mit.edu/18-06S05
RU-vid Playlist: • MIT 18.06 Linear Algeb...
18. Properties of Determinants
License: Creative Commons BY-NC-SA
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5 май 2009

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@PhucLe-qs7nx 2 года назад

00:00 det(I) = 1 03:16 det(P) = 1 or -1 07:00 The determinant is linear in **each** row. 11:34 2 equal rows => determinant = 0. 14:34 det(A) = det(U). 19:00 Row of 0s, determinant = 0. 22:20 det(U) = product of pivots 28:30 det(A) = 0 A is singular 37:40 det(AB) = det(A) * det(B) 41:40 det(A^T) = det(A) 47:00 Odd / Even number of row exchanges

@stearin1978 2 месяца назад

DEFINITION PROPERTIES: (1) Identity matrix, det= 1: no proof. (2) Rows exchange reverse the sign: no proof. (3) Linearity on rows: no proof. COROLLARY PROPERTIES: (4) 2 equal rows, det = 0: proof based on (2). (5) Basic Gaussian elimination transfromation (i.e. row_k - l*row_i) doesn't change the determinant: proof based (3) and (4). (6) row of 0's, det = 0: proof based on (3) and row of 0's=row_k-row_k. (7) Upper triangular, det (U) = d_1*...*d_n: proof based on elimination process (5) and (1) and (3). (8) det(A) = 0 A is singular (invertible). Proof: approximate, based on Gauss elimination process (A -> U -> D). (9) Homomorphism of groups property: det(AB) = det(AB): NO PROOF!!!. So... Integer power works: det(A^(-1)) = det(A)^-1, det(A^2) = det(A)^2. Scaler pops out with power of n (Volume property): det(kA) = k^n det(A). (10) Transposition property: det(A')=det(A): proof based LU decomposition. As always honorable Gilbert Strang showed only the nice proofs.

@sepehrs6098 10 лет назад

Gilbert Strang is gifted in two ways. Not only he possesses the knowledge and expertise necessary to be a math professor, but also he has the charisma that encourages people to both listen and enjoy what he is talking about.

@DanielCoutoF 9 лет назад

Sepehr S Totally agree with you. I feel so lucky to live in an era that allows common people like me to learn directly from such amazing and taletend people like pf G. Strang. Sheers from Brazil

@Nakameguro97 9 лет назад

QING XIE Agreed. Herbert Gross on Calculus and Gilbert Strang on Linear Algebra are equally amazing.

@nofiltariq1726 4 года назад

exactly

@tolunaytemiz 3 года назад

disagree, the former has to do with a lot of work not with giftedness.

@henrypeterson8497 2 года назад

Watching 3b1b's series and then this one paints such a rich picture of the connections between the symbolic and geometric interpretations for these ideas. So fascinating to see the differences in their approach, yet so illuminating to piece together the equivalences and build a deeper conceptual understanding.

@Mark-nm9sm 10 месяцев назад

Yes!!!! literally , also i did a bit of khan academy prior to this . Im beggining to like math and not be so immensely intimidated by it. All 3 teachings have so far been top notch

@Soapluvva 5 лет назад

Greetings from New York City! Prof. Strang was my Linear Algebra 18.700 professor during my sophomore year at MIT in Spring 1973. I loved his teaching style, did well on the tests, and received an “A” in the class.

@thangibleword6854 5 лет назад

and what are you doing here? Revise

@GavinoFelix 10 лет назад

I love when he's building up to property 3. You can tell how excited he's getting (adjusting his hair, taking little pauses). Very cute

@vlogsofanundergrad2034 4 года назад

40:30 The 3B1B perspective of areas and volumes for determinants.

@bfl9075 2 года назад

Determinant is the first chapter in some textbooks, which makes students lost in linear algebra. Dr. Glibert made everything easy to understand, thank you!

@DaWanderer 14 лет назад

i didn't learn anything about linear algebra at my own university. then I found these lectures from MIT. this stuff is GOLD!!

@technoshrink 9 лет назад

Another way to prove that the determinant of a matrix with a zero row is zero would be to add one of the other rows to that row. That would create a duplicate row, which he proved the determinant of was zero.

@ozzyfromspace 4 года назад

clever

@integralboi2900 4 года назад

Big brain

6 лет назад

This is the best introduction to determinants, that I have seen so far.

@martintoilet5887 4 года назад

Same

@Gritpawa 3 года назад

#facts

@therealsachin 2 года назад

You must watch 3blue1brown’s video on linear algebra.

@hubomba 2 года назад

I was in the Sheldon Axler camp that linear algebra should be taught abstractly with focus given to linear transformations from an almost exclusively algebraic point of view and hold off on determinants until the last possible second. Strang's lectures change all of that, these are the best lectures on linear algebra taught from the angle of matrix algebra. Both approaches are valuable but Strang really nails it. Deriving the determinant from properties like this provides excellent motivation versus just jotting the nasty formula down, it also helps with mathematical maturity since it forces understanding by the student.

@abramcz Год назад

I am so very grateful that MIT decided to make these courses available on the web. A very generous and civic-minded thing to do. The fact that a middle-IQ, middle-income person like me, living very far from Massachusetts, can get this level of teaching for free on my computer is almost too good to be true. Thank you, MIT!

@karthik3685 2 года назад

I learnt these properties in my 11th grade. Had no intuition for it, nor any particular interest. I knew the rules, knew how to apply them, knew how to solve 'tricky' problems. This is just sublime! So lovely! I am totally in awe.

@sur1kor 13 лет назад

I am so addicted to these lectures.. Thanks MIT and Prof Gilbert Strang.

@kinghassy334 2 года назад

I am currently a software engineer in 3rd year. I watched a couple of these lectures back in first year when I was taking linear algebra and found it extremely confusing because that was my introduction and these lectures expect you to know the basics. Now that I know the basics and am currently reviewing what I've already been introduced to, these lectures are super insightful.

@aleksybalazinski 3 года назад

A quick observation for 39:45. We can also prove that det(2A)=2^n det(A) by noticing that 2A=2IA=DA, where D is a diagonal matrix with twos down the main diagonal. Then, det(2A)=(by 9)=det(D)det(A)=(by 7)=2^n det(A).

@jerryzack 3 года назад

Yeah, bro, you wrote what i want to post.

@kanikabagree1084 4 года назад

You are the best maths professor i've come accross thankyou so much Dr Gilbert Strang you're a blessing for all the students who struggle with linear algebra

@vozzen 7 лет назад

One important thing determinant says about a matrix is how much the volume/area/length is changing. Everything become much clearer with determinants when you learn this fact. There exist very good videos on RU-vid about it, where you can see it in action.

@btsjiminface 5 лет назад

Oh, that's explains the meaning behind how we know from previous lectures that for projection matrix P, P^n = P, which means that det(P^n) = (det(P))^n is true only if det(P) = 0. The multiplying the projection matrix to b multiple times does not change the length of the projection of b.

@thedailyepochs338 3 года назад

especially 3 blue 1 brown , the image from his video just came to my head

@freeeagle6074 2 года назад

This is taught in Lecture 20.

@chuntelee4371 4 года назад

This is the most correct way to learn linear algebra. Establish direct sense but not be buried by thousands of definition and proof.

@Viggen66 Год назад

Thanks so much Prof Gilbert Strang, I wasn't understanding Determinant usage, and now I fully understand owing to the proprieties, which makes it so simple. you have just a gifted skill to teach so well and clear, before answering the questions, putting the shoes of a student, to think of a possible solution and applying what u have just teach, and applying to problem solving instead of mechanizing meaningless math rules.

@drgothmania 2 года назад

I've never had a chance to learn determinants this way. It was always that teachers gave out a bunch of formulas and methods and made students memorize without further explanations. Thanks for walking us step by step through this wonderful concept, Prof. Strang.

@cache-re8if 3 дня назад

21:43 "Your idea is better" - very humble!

@hj-core 10 месяцев назад

The way Professor Gilbert teaches the determinant is just amazing!

@swapnils6902 5 лет назад

While in high school (in India), I used to hate matrices, determinants, and vectors. They taught it like they were just a bunch of mindless, random calculations. Prof. Strang gives meaning to all of them and linear algebra has suddenly become wayyyy more interesting!

@rinkaghosh7961 3 года назад

How valuable these lectures are !!! Kinda Addicted !!!! 🙏 Thank you very much Prof. Strang and MIT 🙏

@Alex-bc3tt 2 года назад

This deserves to be in the Guinness book of records as the best introduction to determinants 🙏🙏🙏🙏

@mrpotatohed4 3 года назад

Never learned determinants like this, always just given the formula and the applications. Very enlightening.

@replicaacliper 4 года назад

That was actually the coolest introduction to determinants ive ever seen and will ever see probably. Hopefully he brings in the physical intuition later

@supersnowva6717 Год назад

Everything is so clearly explained and laid out! Thank you so much Prof. Strang!

@sumitgupta6905 3 года назад

Best way to teach determinants. I used to worry about how this is the worst part of linear algebra since it involved a big formula that was thrown to me. I loved the intuition about the property 9 about volumes of n dimensional cubes. Never thought determinants would get me this excited. Long Live Prof. Strang. Thankyou MIT.

@veenashalini4904 4 года назад

I blessed to see the prof.Gilbert strang lecture. Very thankful to u.

@kavanavvasishta4692 5 лет назад

Love you Professor. You are such an adorable person and a great great teacher!

@karimkaan8700 6 лет назад

Thanks to you we really start to see what s going on in Algebra

@yevonnaelandrew9553 3 года назад

The best explanation on the determinant of matrix ever! Thank you.

@quirkyquester 4 года назад

ayeee back here again!!! This is a great lecture! amazing details about the properties and proofs. I was just wondering how the hell det A = ad-bc. can't stop thinking about it. And here we areeee. Professor Strang proved it!! Thank you Professor Strang and MIT OCW!! U are the best!

@DareDefyMe88 13 лет назад

These videos are amazing for test review. My linear algebra teacher is awesome but these videos are nice to watch since hopefully I already know everything going on.

@shailenderkumar7315 5 месяцев назад

Sir, thank you for your inspirational lectures, your style of delivery really motivates us to appreciate the structure and beauty of mathematics developed in a step by step way

@thomasrad6296 6 лет назад

33:36 "That's what she said."

@shakirulislam2477 2 года назад

I learned more about determinants within the first 5 minutes of this video than I did in my 3 hours of lectures on the topic so far.

@ouahidlakhal5934 2 года назад

Great teacher. thanks you MIT for the high quality courses you share.

@salehjamsaljames 13 лет назад

my prof. can come and learn from this prof.

@adhoax3521 4 года назад

Mine can't even learn since he's dumb . they are full of dumbness.

@dataInterpretations 3 года назад

haaa haaa

@ndonyosoko5680 3 года назад

Especially those professors who read math slides derived from textbooks.

@trevandrea8909 4 месяца назад

😭 😭 😭 @@ndonyosoko5680

@morganjones7428 3 года назад

What an absolutely incredible lecture!!!

@vonfred54 14 лет назад

the most interesting algebra ever. thank you professor!!

@wduan1225 9 месяцев назад

this is 100x clear than the linear algebra course i took back in college, good teach does make a difference

@cafe-tomate 6 месяцев назад

For rule 6, take the matrix 2×2 {0,0, c, d} and write the first 0=c-c and the second zero=d-d

@lovvyparhar393 4 года назад

The best determinant lecture I had!

@nota2938 Год назад

In the last bit, there lies the fact the alternating group A_n could be viewed as being a normal subgroup of index 2 in the corresponding permutation group S_n. Algebra is a fun topic.

@tongeason1235 Год назад

I start learning ordinary differential equation and Laplace transforms and I found the method of teaching was decent and clear( better than many uni professor)

@ahmedkarakaci6118 4 года назад

Thank you Professor Strang.

@afrakilic5672 8 месяцев назад

I just enjoy to watch these videos... so much...

@muratbayraktar4114 3 года назад

one way to prove property 4 is to use property 2: ( a b; c d) = ( tc td; c d) then the determinant is t* (cd-cd) which is 0. I love the way prof. Strang teaches it's inspiring

@anhkhoangothe6793 5 лет назад

The definition is just brilliant

@marverickbin 6 лет назад

best det class i ever had.

@juancruzparada8690 Год назад

absoluty beautyfull, im loving those clases, congrats from Argentina, keep giving us those amazing clases, plz

@DanielSColao 4 года назад

Fantastic lecture!

@Hepichack 9 лет назад

It is just fantastic !!! Thank you !!!!

@Meritjamtsho 12 лет назад

Man! you are an inspiring mathematician. If i become one, full credit to you.....

@danieljulian4676 5 лет назад

~22:10 "now I have to get serious" so, what was all that other stuff?

@ja2smart92 13 лет назад

He is really good...thx for uploading this

@julietteaqui 4 года назад

GS: "the determinant of an upper triangular matrix would be just (d1) times (d2) times ...*hands rotate* (dn)." me: *writes (dnd)*

@bhatiaayush11 Год назад

18:23 "I'm ready for the kill", amazing!

@type144 13 лет назад

thanks a lot, professor

@FabledNarrative 5 лет назад

The way he says "Kill" makes him sound serious @ 25:56

@pd1769 5 лет назад

He's a serial killer Ok bad joke sorry

@danishji2172 Год назад

Idk if anyone else felt this way but this man has the charisma of a fatherly figure. It is hard not to like him, and a lot.

@yoitsrob 2 года назад

Great insights here

@RiSuJiCh 11 лет назад

This works for me. Thanks

@spacesher3135 3 года назад

"Now how do I prove that?" Me: Gilbert bruh, chill not necessary. I trust you with my whole life.

@jaswanthbyjus1288 3 года назад

Great guy in the planet

@jaswanthbyjus1288 3 года назад

Thank you MIT

@jadenmax679 4 года назад

very informative, thanks.

@greekman234 14 лет назад

Dterminants is my fav part of linear Algebra xD

@RaviRahulKumarShah 5 лет назад

wow ! That's Beautiful !!!

@miyamotomusashi4556 2 года назад

Ooooooooooooooooh my god this is soooooo beautiful. Thank you.

@Labroidas 3 года назад

I'm really thankful to the people going through the trouble of making these subtitles, but in this video the subtitles were so full of errors of all kinds that it was really irritating :( I had to turn them off eventually because they distracted me so much.

@dAntony1 2 года назад

He’s more effective with that one HUGE piece of chalk than my professor with her ipad and Zoom. She should just assign this as the lectures and be done with it.

@rowechenzhong8950 5 лет назад

800th upvote. Very good video!

@aman201156 3 месяца назад

Great Professor

@LennyLeonard85 13 лет назад

@seisdoesmatter The chalk's really awesome. Looks thicker than ordinary chalk. It seems a bit like the chalk kids use to paint on the pavement!

@alijoueizadeh8477 5 лет назад

Thank you.

@giri2344 Год назад

Thank you sir🙏🙏🙏

@rampabayir 2 года назад

all the properties of determinants are like his little babies for Prof.Strang. He cants choose one over the other, they are all key properties :).

@serkanvai 13 лет назад

@slatz20 he explains that if you multiply only one row. so he did it correct.

@adhoax3521 4 года назад

Can anyone tell why those three properties exist and why only and specifically them?

@D3tyHuff 6 лет назад

Determinant has been determined. Thank you MIT!

@publicanimal 13 лет назад

Same comment as everyone else. This was a much better explanation than my professor gave.

@nioncao 2 года назад

Amazing!

@ZYau-lc5ql Год назад

I really want to go to MIT😍

@user-oh1dy7hk8z 9 месяцев назад

don't know what he meant in last minute about 'Odd / Even number of row exchanges', cause when we do seven row exchanges and then ten exchanges, that's seventeen exchanges in total, hence sign does change, right? just don't get it😢hope somebody could help me please🙏

@karimkaan8700 6 лет назад

I can see why American people are by far well educated . HAT OFF TO YOU

@mikecohen5887 4 года назад

For property 3b should the top row be (a+a' b) --> (a b) + (a' b)

@zizo-ve8ib 2 года назад

How do you prove the single row linearity of a determinant, I mean how is in matrix A + matrix B we sum ALL rows but in det(A) + det(B) we don't?

@ThePositiev3x 9 лет назад

At 45:30 he describes the "L" matrix as "Lower triangular matrix with 1s on the diagonal". In this case "L" is not exactly lower triangular but a special form of lower triangular matrix. Isn't it? Because I think the diagonal of lower triangular matrix doesn't have to consist of 1s ...

@txtlu6898 8 лет назад

+ThePositiev3x A can be factored to LU in which L is lower matrix with 1's, check his book in page 97.

@NuncNuncNuncNunc 8 лет назад

+ThePositiev3x, from your question I assume you already understand this so this clarification is really for others Here it would not matter if the diagonals were all 1s or not. L and L' have the same trace and all zeros below or above the diagonal so they have the same determinant. Making L unit triangular forces the LU factorization to be unique. The particular form of L here, Prof. Strang shows in earlier lectures, comes from combining the inverses of elimination matrices that have 1 on the diagonal. As a practical matter, using 1s in the diagonal makes showing the linear algebra clearer by eliminating a lot of arithmetic.