What's the big idea of Linear Algebra? **Course Intro**

This is the start of a one semester university level course on Linear Algebra that emphasizes both conceptual understanding as well as procedural fluency with the techniques of Linear Algebra. In this video, we get to see just the beginning of some of the big ideas of Linear Algebra.
FULL PLAYLIST: • Linear Algebra (Full C...
Linear Algebra is a story told in two worlds, one geometric and one algebraic. The interplay between the two is beautiful and powerful. In this video we introduce the notion of a linear transformation and see how this simplification makes our lives far easier.
Now it's your turn:
1) Summarize the big idea of this video in your own words
2) Write down anything you are unsure about to think about later
3) What questions for the future do you have? Where are we going with this content?
4) Can you come up with your own sample test problem on this material? Solve it!
Learning mathematics is best done by actually DOING mathematics. A video like this can only ever be a starting point. I might show you the basic ideas, definitions, formulas, and examples, but to truly master calculus means that you have to spend time - a lot of time! - sitting down and trying problems yourself, asking questions, and thinking about mathematics. So before you go on to the next video, pause and go THINK.
This video is part of a Linear Algebra course taught by Dr. Trefor Bazett
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[@gastondupertuis8009]

First of all sorry if my English isn't perfect. I'm from Argentina. I'm near 42 years old and started this year to study Electronic Engineering. Is never to late :) This is my first comment ever in a KZbin video and I want to encourage you to keep posting more videos because they are very good. Great explanations. I hope more people could discover your channel. Thank you very much and keep going!

[@realking4918]

I'm 32 and I agree!

[@izgit]

@@realking4918 I'm 22 and I agree!

[@chrisedwards3640]

@@izgit I'm 12 and I agree!

[@TheOtherBradBird]

Howdy from Louisiana, Argentina. I agree, this man just has a knack for unpacking and presenting. Also, your English is perfectly fine, stellar for a second language speaker. Only hiccups were that we'd probably use "Electrical" rather than Electronic and your tense matching in your second-to-last sentence is a little off. Worst case scenario is mild confusion, which is wonderful. Good job to you and keep it up.

[@gastondupertuis8009]

@@TheOtherBradBird Thank you for your feedback!

[@ryantierney629]

Currently taking linear algebra at Harvard and studying for my midterm, your explanations are coming in clutch.

[@DrTrefor]

Thank you and good luck!! When I was first learning linear algebra well over a decade ago, I used videos from MIT. Feels full circle to have harvard students watching now:)

[@andrewsaul9451]

@@DrTrefor Was that performed by the excellent and entertaining Prof Gil Strang at MIT?

[@urmomsh0use]

Does it help your ego to tell the comment section you are at Harvard?

[@sudarshann7194]

@@urmomsh0use and does it help your jealousy to cool abit by putting-out a taught of you that he's egoistic being in harvard . Oh Comeon bro , that's fair of him . Thing no. 1 is, he wanted to make a point out-of saying he's studying at Harvard . ( That point is, he found similar explanation in this video and in there ) Thing no.2 is , its even ok if he's egoistic , bcz as we know it's one of the best institutes in the world , which is very hard to get into such institutes and he has able to do it by his hardwork . You need to respect atleast that .

[@cringetopia6508]

@@sudarshann7194 hippity hoppity rakesh

[@ModiSaurabhMehulBEE]

For those who are not able to feel linear algebra even after studying it somewhere(in college or by some book), do one thing just binge-watch videos up to 50 videos consecutively. It would take about 4-6 hours and then you will know what is the motivation behind the different concepts. I did it and one takeaway that I drew is that linear algebra is a study to develop types of algorithm to visualize the vectors and systems of linear equations. It's very useful to develop useful computational algorithms as it makes the job more modular and recursive for computers to apprehend. It might sound vague but think of it after watching videos and you will get ideas of making several algorithms to solve problems like making a program to find the solution to a system of equations, making a program to find the rotational coordinates in a system, etc.

[@philipjuras5360]

When you say 50 videos, do you mean 50 of the videos from the playlist this creator made?

[@BabaBoee5198]

Thanks I really appreciate it

[@BabaBoee5198]

​@@philipjuras5360 I guess. I also binge watched Dr trefors videos altogether and just when I completed the derivative video, it all just clicked and oh man the sense of satisfaction, words arent enough

[@michellechal4669]

you have NO IDEA HOW HELPFUL YOUR VIDEOS ARE

[@WhySoSerious6965]

People like you are what makes the internet such a blessing.

[@ZeerobForever]

I have been taught linear algebra at school a long time ago. Matrix operations, multiplying matrices, etc. But is was never really clear to me what these numbers actually meant. I saw your 13 minute video, and now it is finally clear to me what a matrix actually stands for. Thanks a lot!

[@MrMentholSlim]

30 year old burn out going back to school. your videos are impossibly helpful for understanding stuff in a "less formal" way then taught in class.

[@ninaddongare6514]

All my school I thought math is such a waste of time with no use. I wish I had a teacher this passionate, I probably had a different career at this time. Anyway, better late than never.

[@GautamKhatter]

This is the video that solved the confusion in my mind about linear algebra that has been stuck in my brain for days now. Thank you for explaining this with such easy examples. Would love to see something like this for the higher dimensions.

[@skipintro9988]

I studied linear algebra at university; but actually learned from this playlist. Unarguably it is the best place to learn linear algebra on the internet. I wish he was my math teacher at high school.

[@Ianwong33]

Professor Bazett, I was one of your students in MAT223 (The Linear Algebra course) when u were still in UofT. I did not expect I would see you again on youtube. Thanks for your teaching anyway! Keep up the good work

[@DrTrefor]

Woah, that’s cool! That was three universities ago for me!

[@Ianwong33]

@@DrTrefor yes, time flies. I graduated 4 years ago and I m currently working as a math tutor in a high school now. That’s why I am here to go over the linear algebra stuffs again. Thanks for making some great contents and it actually helps a lot !

[@michelledlr4125]

im gonna study engeniering chemistry and i think i got pretty clear the idea of what linear algebra is

[@FirstLast-gk6lg]

Awesome! Never got past Algebra 2 in high school and about to start Computer Science next week lol so I am binging Linear Algebra stuff and this content is exactly what I was looking for! About to binge watch that playlist you had at the end of the video, thank you.

[@nicholaschan2518]

You’re a fantastic teacher.. I wish I found this video sooner! Cheers sir 🙏

[@user-pk4hn1uz1k]

your explanation of how a matrix relates to individual vectors with the graphics was great

[@daisyg509]

My god. I'm in linear right now and having trouble understanding why we're doing this math but in a matter of minutes it makes more sense to me. Thank you.

[@DrTrefor]

Glad it helped!

[@SaimKhan-xj5um]

you are awesome,please dont stop posting

[@picklerick8844]

At 11:55, the inverse matrix should have been: [1/3 1/3 -1/3 2/3]

[@AbuSayed-er9vs]

Mind blowing!!!Your linear algebra videos are really really good to understand to get a clear picture.

[@sierrap6478]

Taking Linear Algebra in the Fall (a little over 2 weeks from now). Hoping to get a jump start and calm my anxiety about the subject with these videos. I suck at math so hopefully this video (and the videos following) will help

[@T-The-K]

im having the exact same issue, im super anxious. But i only have 1 week. How did it go for you? and how far did you get with the playlist?

[@sydneeicloud247]

how did it go? I am about to take it this spring!

[@pjakobsen]

Outstanding intuition -really excited about this course. Thank you Trefor!

[@prateeksenapati2152]

Amazing explanation Professor Bazett! Got me interested in Linear Algebra. Those visualizations help a lot : )

[@VishalJangid1]

Are bhai what are the odds tu yaha dikh gaya mai jamnagar me tha yaad hai? :P

[@abhirajarora7631]

@@VishalJangid1 Coincidence moment 👌

[@bryannac8541]

This helped me soooooo much! For a few years now I have struggled with what linear algebra actually is. You cleared it up so much in such a concise way; thank you!

[@mr.wolfmaster2892]

Anyone else here for fun?

[@liliiamaslarova1840]

Just started my course today. Thank you for your effort! Excited to report along the way how I feel about the class:) Thanks again!

[@EyeIn_The_Sky]

For years the fact that although the functions is read "F of X" (i.e it is read left to right) the actual application is right to left

[@NothingMaster]

There is no ‘magic’ behind the algebraic and the geometric aspects of Linear Algebra, once you recognize the fact that in modern mathematics we prefer the approach using the notion of vector spaces (and their associated axioms). Then the linear maps (or the liner transformations, if you will) are simply the mappings between the vector spaces that preserve the vector space structure.

[@davidsandham]

Great video, despite the mistake at the 11:53, where in the inverse matrix the leading diagonal numbers should be flipped.

[@maxpercer7119]

I love this overview, nice to motivate the subject.

[@oes2546]

I’m guessing maybe not since no-one has mentioned it, but isn’t the inverted matrix actually {{(1/3), (1/3)}, {-(1/3), (2/3)}}?

[@TUMENG-TSUNGF]

I noticed it too! Thank you for confirming my doubt!

[@paulmesler5715]

That was probably the single best explanation I have ever heard. Thank you.

[@ManojKumar-cj7oj]

I'm gonna start this playlist right now , Keeeeeeeeep making these please ❤️❤️❤️

[@malcolmvernon6808]

Man i like your teaching style... you give strong motivations for everything. For example... khan academy tells you diagonalization with no motivation... you first state why its awesome and then go into it. I also like your editing style... you tend to skip the video time in writing something... put it down and explaine what it means. Long story short.. i would like to invent good things some day. And I hate having to learn a concept and it be vauge i dont like it.... I like your style. I find it easier to keep track of things in your videos. Im probably going to end up watching everything you ever uploaded. no lie

[@John231984]

I just had to "Start" (At 40 mind you) taking this for a college course. I'm not very good at it, so thank you so much for posting.

[@MrMusicM67]

Dr. Bazett, this was so captivating! What software do you use for your graphics?

[@liamgauvreau]

I’ve been watching vids from your channel for so long and just today found out that you’re a professor at UofT. I’m in grade 12 and I got accepted into the Physical and Mathematical Sciences at St.George, definitely looking forward to taking any courses you teach!

[@DrTrefor]

Cool! Although sadly I USED to be a prof at UofT but am now at UVic so you can't have me in person, but I have lots of great colleagues at UofT and I think even a couple assigned my vids during the pandemic. You'll have so much fun though, I'm happy for you:)

[@liamgauvreau]

@@DrTrefor Aw shoot I should’ve time travelled and graduated a few years earlier! Thank you so much for the comment, take care:)

[@tonytucker8651]

@@DrTrefor Hello, in this video, you say the red vector is two to the right. I am confused because it appears to be one to the right. How are you determining it is two to the right?

[@roobear5357]

@@tonytucker8651 It is two to the right based on the original rectangular grid… The rectangular grid unfortunately wasn’t included when he drew the new red vector… The red vector is one to the right based on the new nonrectangular blue grid system

[@tonytucker8651]

@@roobear5357 He started with a square 1 by 1 unit when he turned it and stretched it to a rectangular shape. It still appears to be one to the right. I do not see the transformation to two to the right.

[@ayae.muhammed7305]

You are one of my best instructors ! Thanks alot !

[@sergiolucas38]

Great video, i intend to take the path of a STEM student and i hope that linear algebra help me out with this :)

[@TheFlanagan11]

Great video Trefor. You're such a legend!!!

[@daoudatraore930]

Thank you very much Trefor, i really enjoy watching your videos, they help a lot in my studies :)

[@BIMALPARAJULIBDS]

This man is magical

[@destinydiaz9791]

I took linear algebra 5 years ago and failed it twice. I'm brushing up with your videos and hope to pass one day :)

[@Lemurai]

What part did you not do well on? I found Linear Algebra to be a little easier than differential equations and calc 1-3, I’m not that smart, trust me lol! Have you considered taking it at a community college? They usually do a better job of teaching mathematics imo, the worst mistake I made was taking calc 2-3 at my university, I should have stayed back and finished it all in the community college system, smaller classes means more time with the students.

[@be_present_now]

Thank you @DrTrefor for this amazing playlist on LA. Just one thing to add here - Singular Value Decomposition (SVD) is missing in this playlist professor. Would love your explanation on this important topic!

[@andrewsaul9451]

I love these lectures. Just wondering if @11.58 your inverse matrix is correct? Should the leading diagonal entries be switched?

[@charlesadami8248]

I was the 43rd comment. 42 is an important number. 10^42 means everything to me. The ratio of the electromagnetic force to the gravitational force and the diameter of the universe to the diameter of a proton. 42 is god. 42 is life. 42 is the answer. But...what am I, as comment # 43?

[@robharwood3538]

You are: The *successor* of 42. Something to be proud of! It's like: "What am I? The mere remnants of extinct stars that exploded and produced all the matter that the Earth, and thus all known life, is made of. 😥" "Dude, you just said it yourself. You're literally a conscious being, born from supernovae and even the Big Bang itself. You are the universe, contemplating its own existence. Stars *died* so that you could *live!* What more do you want???" 😉😊

[@prempresents8776]

This course helps in machine learning because I wanted to learn linear algebra for ml

[@stormsm911]

His shirt says: Sleep, Eat Pie, Repeat xD

[@kolo6518]

My life right now

[@fajarsyaban5336]

Lmao

[@tirthsolanki90]

It's Pi.

[@Chris77Rocks77]

I just had an epiphany because of this video. I realized that when you look at these 2-d vector translations through a 3-dimensional lens, it's equivalent to moving your eyes (head) around the object and looking from different perspectives!! Someone please correct me if I'm wrong

[@Lemurai]

I’m just coming back to this after a loooooong break. After I finished my Chem E degree, I was so burnt out on math I took a job as a waste management supervisor lol😂 because I had the engineering background they tried vehemently to integrate me into reclamation/processing with the bio chemists, but I wasn’t having none of it, I didn’t even care about it the pay gap lol! Math is great but it’ll burn you, looking at it on days, weeks & years on end.

[@tyrmyrmidon2846]

There are spaces opening up in my brain

[@r_majka]

Hi. I am studying IT engineer and this videos are very useful. Thank you

[@DrTrefor]

Thank you, glad it’s helping!

[@nl5h]

the issue i had with linear algebra is it stressed linear independence vs dependence. i get the aspect of how that effect a transformation NOW but it wasnt emphasized enough in a practical application for me back then. this was a 4 week course at ucla (coupled with differantial equations)

[@mikarpman]

Overall this was a very well done and clear video. But I am stunned that there is a glaring error in it that was not corrected. The inverse matrix is incorrect. I have seen other KZbin videos that have pointed out mistakes after the fact, but for some reason this one does not. It does not inspire a lot of confidence that the channel does not seem to care enough to correct its own mistakes.

[@duydangdroid]

super informative. going to watch every video on playlist.

[@DrTrefor]

See you at the end!

[@jordonwu5426]

Which book you recommed for Linear Algebra? thanks

[@DrTrefor]

It REALLY depends on the level of the course and how proof-centric it is. One I’ve used many times is Lay.

[@pgfinna]

I......understood that.......

[@user-rc6xt1fi5n]

The explanation was brilliant. I really appreciate it!

[@richardojuan1484]

Great work man!! love the coin analogy

[@dolphinsatsunset1]

So in a sense the origin of the x-y-z cartesian space is kind of like Polaris and the other points move around that centered point when they translate to other lines

[@spacebatstuckonearth8888]

5:53 is just seconds way before my brain reboot

[@MaggieGaba-doo-doo]

Thank you! Love your videos. Mathematics is so magical. 🌟

[@lakshmi4554]

Thanks for your videos. Could you please do a video on Probability distribution?

[@khaledfarrag9754]

Could you make a series of type of geometry and mechanics also

[@EyeIn_The_Sky]

Can you tell us what this actually tells as rather than it being purely an exercise in process? Like some practical application (or even advanced application) on what it shows us or proves or answers that we can use to make or solve things?

[@faizanmohsin3685]

Just rotate your mobile screen for a picture or video. The center of picture remain center but other points and length stretch out to make picture bigger in size. Think about this you will get your answer

[@continnum_radhe-radhe]

This is awesome 🔥

[@chandankar5032]

Do you know any software where I can see this transformation of planes?

[@MeistroJB]

Right. Clarification simplifies. Something like Gaussian simultaneous. Really does, since nobody wants to to it with 'regular' algebra... right..? Especially in six or eight dims. Right, Wolfram?...

[@ninadeck3577]

I don't get how those equations came from those points. I mean the -1 is the x coordinate of the second point, why is it multiplied to y_in.

[@cicerohen]

Your explanations are really great. Congrats. :D

[@Savedbygrace952]

does that an inverse of the first matrix? [2/3 1/3; -1/3 1/3] is not the inverse. do I miss the point of his explanation?

[@neocrz]

very interesting

[@stevewall7044]

Thank you for this video.

[@Levelup383]

Thanks for the lecture ❤️

[@user-dg7pv3jy5d]

Just a quick question, When red was at coordinated of 0,1 and it shift its coordinated at 6.20 sec, it went 1 up in y axix, you are stating that it went 2 to the right and 1 up, i understand that figuer move upward but it didnt move the right side, so shouldnt it be 1,1 rather than 2,1 ?

[@Lluks4]

Oh... that makes sense. Thank you!

[@ankurpatel8679]

Who is indian here 👍👍👍👍👍

[@cursedtosufferGRIMM]

Not me, but I'm hungarian, and I love indian food and wanted to learn hindi, but I gave up

[@leviberlin4520]

Me!!!!! (I am actually white)

[@iamcase1245]

What does it matter who's Indian?

[@ankurpatel8679]

@@iamcase1245 indian means not matter india is a country understand

[@iamcase1245]

@@ankurpatel8679 what language are you typing in? No in English I understood nothing.

[@kharcasse5303]

Great video 👍

[@lukebrbot1791]

1 done 81 to go, thank you very much for making these😅

[@nujchareehaswell4701]

This is great - thank you so much. I saw there are 4 questions you listed out - please I wonder if you posted the answers anywhere I would like to check my understanding. Thank you :)

[@robharwood3538]

@@DrTrefor Hmm, as this is partly a video on the motivations behind Lin. Alg., may I suggest that Nujcharee's request is perhaps more representative of a real human *need* for being able to verify that one is headed in the right direction -- particularly in a timely way, such that the verification can serve as *positive motivational reinforcement* -- than a mere request for an I.O.U. for the promised 'answers' (read 'the much-needed positive reinforcement') at some vaguely-specified time 'in the future' (as opposed to 'right after I've just worked through it on my own, when said positive reinforcement is most needed')? Really, what's the harm for posting at least a link to where one can find the answers, for those who've actually done the work and could use the motivational boost? ("Yay! I got it right! 🥳") Why make them wait, especially with such vagueness about _when_ the feedback will come? ("Oh... well... I guess I'll have to wait a few days... or maybe weeks... or who knows how long... before I'll be able to verify my current understanding. 🙁 Hopefully I'll be able to stay motivated ... somehow ... to stick with it until then ... whenever that may be ... if ever.... 😢") Just a suggestion, but based on my own experience with lack of motivation around Linear Algebra specifically (see my other comment on the main video). Personally, I find that concrete, timely feedback is an essential component to learning motivation, and I believe (from my layperson's understanding of some of the science) that it's not just me: The entire dopamine feedback system in the brain (the basis for forming habitual behaviours, and one of the essential components of learning (as learning requires learning-habits)), is dependent on receiving feedback from effortful actions, and particularly on the _timing* of such feedback -- too long, and the feedback becomes essentially useless, too dissociated from the effortful action. Anyway, just my two cents. 😊

[@RetroGamingClashOfClans]

2:18 Microsoft logo function

[@BhaveshGahlot-oi5pi]

i think u forgot to make a video on linear operator welcome your new subscriber from india

[@hhkkv2]

Simply awesome !

[@gajkrishna9989]

A blessed passionate Teacher best of the best, Thank you 🙏

[@DrTrefor]

Thank you! 😃

[@dolphinsatsunset1]

the magnitude of [1 0] should be 1. right? so i don't understand why the transformation of that vector i.e. [1 0] should be [2 1]. Am I missing something? Please help me understand this. Thanks. btw the magnitude of [2 1] is sqrt(5)

[@joaomattos9271]

Thank you very much!!!!!

[@lisali5719]

great video

[@bill5367]

Great videos. I think he mentions the word text or textbook somewhere. Does anyone know what book that may be?

[@OfferoC]

This is awesome. Thank you so much.

[@georgeschlumberger9814]

What if i said i like your videos more than 3B1B?

[@georgeschlumberger9814]

@@DrTrefor i wish there was a physics channel as good as yours🤔

[@hamzamahmood8980]

6:40 How are we sure about where the red and yellow vectors transformed?

[@00zytes]

Sound is clipping, turn down the mic gain

[@nicklayon]

Any recommendations for a source of problem sets or textbook to accompany your videos?

[@Galaxia53]

Am I supposed to learn everything related to mathematics before learning linear algabra

[@DrTrefor]

nope, it requires basically zero prerequisites.

[@dogwithwigwamz.7320]

I think its about time linear algebraicisians decided how many dimensions a point occupies and how many dimensions a line occupies. Once this is resolved I think I could make some progress with the rest of what might come.Some say a line is 2D and that a plane is 3D whilst others say a line is D and a plane is 2D.

[@dogwithwigwamz.7320]

Hi, Trefor Bazett. You`ll forgive my naivety. I am new to maths and see only that `Wiki..` defines a one dimensional object as a line whilst `Math is Fun` defines a line segment as starting at ( x`,y`) and ending at ( x*, y* ). Wiki also saying that the dimension of a mathematical space is defined as the minimum number of coordinates required to locate any point within it. The Wikipedia definition makes more sense to me, so I`ll stick with it for now.

[@onelinerpreparationmaths]

if i have a linear operator T, then how we can prove image of a subspace of a vector space is vector space , kindly show

[@vaginalarthritis1753]

I know this isn't related too much, but I'm having some issues with a simple vector problem. Well, 1 aspect of it atleast...the angle Assume the following fact: If an airplane flies with velocity *V(a)* relative to the air and the velocity of the wind is *W* then the velocity vector of the plane relative to the ground is *V(g) = W + V(a)*. Now suppose that the Wind is blowing from the North-East at 50mi/h and that the pilot wishes to fly due East at 500mi/h. What should the plane's apparent velocity vector *V(a)* be? My issue with this problem, isn't that I didn't get an answer at all, I got a vector of magnitude 537 approximately which is correct, but I got an angle of around 4 degrees but the text book says that its 86 degrees. I can't find where I went wrong in that calculation. Any help you can offer will be greatly appreciated.

[@selindoga]

Now I understood

[@parkerjones6603]

Why can't you be my prof:(( Mine has handwriting in his own language he created and no patience to go over things

[@DrTrefor]

That sounds super frustrating!

[@arielf9587]

278k+ viewers and no one double checked his work? His inverse transformation is wrong, but I'll let you figure out why 😉

[@DrTrefor]

haha someone once around 100k views noticed this too:D But it is shockingly are for such a simple error:D

[@curtpiazza1688]

Cool intro..love your teaching style$