A simple condition for when the matrix inverse exists | Linear algebra makes sense
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- Опубликовано: 17 май 2024
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This video is about matrix inverses, and in particular, I try to give a bit more intuition for them- rather than just giving you the formula for the determinant, Cramner’s rule, the inverse of 2x2 and 3x3 matrices etc. Along the way, we cover some topics that don’t receive enough attention in linear algebra (at least, they didn’t in my math classes), like the left inverse, and why non square matrices don’t have an inverse. Finally, we will learn an intuitive condition for when the inverse exists.
Homework questions
1. What is the inverse of the following matrix:
1 2
2 5
a)
-1 -2
-2 -5
b)
1 1/2
1/2 1/5
c)
1 -2
-2 5
d)
5 -2
-2 1
2. Prove that, for a square matrix, the left inverse = the inverse
3. Prove the above for the following matrix:
1 2
2 5
by first doing these 2 questions
a) Show that M(1, 0) and M(0,1) for a basis. I.e. the old basis gets mapped to a new basis.
b) Show MLv = v for any vector (this is enough to show the M undoes L, and hence show that the left inverse is equal to the inverse).
To do this, first write v as a linear combo of M(1, 0) and M(0,1).
Hints:
3.
a) is a straightforward if you watch my first linear algebra video: • Vector addition and ba...
b) Sub the expression for v into MLv. Then use linearity!
Q2.
a) (Proving that the basis is always mapped to another basis) This is a bit tricky. I've written out the solution but stop reading at the point you think you know what to do and try it.
Assume v1,...,vn are a basis. Apply M to them to get Mv1,..., Mvn.
They're a basis if they're still linearly independent. But if they were, say:
Mv1= a Mv2+... + d Mvn
Apply L to both sides:
LMv1= L(a Mv2+... + d Mvn)
= a LMv2+... + d LMvn
using LM=identity:
v1= a v2+ ...+ d vn.
But this must be false, since v1,..., vn are linearly indep.
To see it done (SPOILER ALERT) 9:20 in • Matrix Inverses make s...
b) Once you have that, proceed in a very similar way to Q3, part b) 
Наука
Two apologises I need to make
1) for not answering comments last time- I was still in Australia and though I read them all, I just couldn't find time to answer them all.
2) that this video is so late- I actually made a video on this topic a couple of weeks ago, uploaded, then decided I just hated it. I scrapped the entire thing and made it all again (and I'm still not super happy with it).
thanks for uploading
Heyyy u did great! Don't let your perfectionism kinda make u burn out from making awesome videos like this!
For what it's worth, I think this video is great. Keep it up, and maybe try not to stress too much about it! We love your teaching style!
Thank you for the very kind words!
If I remember correctly you were going to study quantum computing. I don't really have almost any formal background in physics, but since I study computer science this topic interest me. Is there any chance of you doing videos about it in the future? I really like your content for the most part, and I would like to see your take on the subject.
Just brilliant! A confession: I'm a maths teacher and I never thought of explaning the reason for the existence of the inverse in terms of information. Thanks for that.
Thanks a lot! There are lots of perspectives that work for this stuff so everyone has a bit of a different preference right? This just happens to be mine.
And a jolly good one it is!
After watching 3blue1brown's video on determinants, the condition that the inverse exists if the determinant is nonzero actually makes perfect sense.
"I had no clue" breaks me but is also hopeful and sweet
Even though I really liked linear algebra, looking back I really understood so little!
Thank you sooo much for this! Your explanation really helps me understand visual what’s going on when you take an inverse.
These videos are so well done and thoughtful.
Thanks very very much! I'm so glad it helped :)
Thank you for all these entertaining and educational videos. I learn a lot by watching these!
Thank you very very much! That means a lot to me :)!
Thank you very much for these videos. I really appreciate the way you explain and it somehow makes me feel more interested in the topic. Is incredible! I love you!
Thank you so much, your videos are absolutely brilliant!
Thank you so much!!
You make learning linear algebra interesting and very intuitive. I don't know it at all, so it's neat to get some insight into what it's all about.
Oh thank you very very much- that’s too kind.
What a long time! You’re back, finally.
Oops, sorry! Thanks for returning though.
Important side note:
The determinant shows the nth dimensional equivalent of area that is inbetween where the identity elememts are after they are sent by the transformation.
What I mean by nth dimensional equivalent of area is that in 1d it is length, in 2d it is area, in 3d it is volume, etc..
Basically, use matrix A on all of your elements, and you'll end up with a sort of parallelagram. If you find the area of that parallelagram, that's the determinant of A.
If the determinant of A is 0, then that means that one of your elements "fell" into the others after using the matrix A. This means that that identity element and everything you can make using it is no longer independent of the other elements. You've dropped down a dimension. More even, if other elements also fell.
As an example, think about a 3d space and choose 3 elements, any 3. Now imagine that a transformation moves those 3 elements somewhere else. Now, imagine those 3 elements again, and think about what would happen if 1 of them were moved somewhere in between the other 2 - on the same plane. That shape has no volume, it's 2d.
Say I have a 2D point (1,0) A and rotate it 90 deg by applying M. Is the determinant the 1/4 of the area of a unit circle?
@@Petch85 No. We are not talking about the area between a single element before and after the transformation. We are talking about all elements after the transformation. If you are in 2d, then you need 2 basis vectors. In this case we can use [1,0] and [0,1]. After the rotation transformation, we get [0,1] and [-1,0]. The area is the square between them. In this case, our area stayed the same, so our determinant is 1.
I should mention that area is measured in terms of the unit area you get with your basis vectors before the transformation. The area will always be within a parallelagram. We got lucky this time and it happened to be a square.
+Phoenix Fire Thanks what makes sense. But now I have to look up the jacobian matrix. I thought that gave you the relative change in "area" between two transformations.
Yes, this is a very good explanation of it :)
The quality content can be helpful~ My day has made great by this~
A simple condition for when the matrix inverse exists | Linear algebra makes sense!
This is such a great idea~ Love this very much.
👍🏻
👀
👍🏻👍🏻
She explained this like Prof. Gilbert Strang. Very intuitive!
This a beautiful example.. In fact, all the mathematically forbidden operations are just about losing information. Division by Zero, rearrangement of Series.
Oh my god, I totally agree, but I've never heard anyone else say it before!! Thank you!
Neither have I heard any one explaining like this. A fan here.
Try adding 0 :). It is not forbidden, gets registered as an operation but changes nothing ... and can give you a propler headache too.
Your videos are so helpful! I love how pretty they look too ☺️ thank you! 💗
I like how you approach the concept of inverse from a more examplified way. I tend to use the same approach when I'm explaining things to others. However around 5:39 when you start to explain thw reason why it doesn't have an inverse, it would have been useful to say something like "imagive you are vector w that came from the tranformation M. But since you could have come from more than one vector, you don't lnow which is you father vector."
I think that making it a little more relatable in that w doesn't know were it came from, in other words insisting on this conundrum that w has would make clearer on the main point you were trying to pass along. I'm daying this because I felt it a little flat on the important parts in your reasoning and let me thinking that I was only able to vatch what you were saying because I had this intuition beforehand.
But perhaps is just my feeling and many people were more attentive and did catch the main point for compleating the reasoning. Anyway kuddos on this series, I think you are making a truly excellent job and I do enjoy a lot your work!
Great video I feel like staying away from the proofs really helped get the high level points across
Thanks very much for the feedback- I fretted about this quite a bit so I’m glad it turned out ok.
All of your videos are amazing, because every thing you are explaining pretty well. please do a video on the planks constant that what would happen if it has large value or more small or positive. and also explain the spin of particles that why some particles have half spin and some have integer. every one just explain half and integer spin but not explaining that what reason behind it.
The spin video would be particularly interesting I think! When I understand it I'll try :)
Planck's constant relates energy to inverse time. That means that energy, which is just a quantity of "change" is related to how fast something can change. This was how Max Planck solved the ultraviolet catastrophe. Prior to this hypothesis, there was no relationship between energy and time, which allowed infinite frequencies with finite energy (noting that frequency is just inverse time). Therefore Planck's constant really pins down the relationship between time and energy for a single quantum, which is why it's such a fundamental constant. Another curious property of Planck's constant is that it has the units of angular momentum. Thus quantized angular momentum changes in integer multiples of h/2pi.
Mind you, changing the value of Planck's constant would be catastrophic, as being a fundamental constant that relates energy to a rate of change, it would have a tremendous ripple effect, from changing the fine structure constant, which changes chemistry, to changing how everything interacts... This is called the fine tuning problem. It seems all the fundamental constants are in the Goldilocks zone of being just right for the universe we experience to exist.
This information analogy is a good one. I didn't learn this stuff like this - not even in my applied maths course.
Thank you :) I didn’t learn it quite that way either, but when I realised you could phrase it this way, it made a lot more sense!
It does indeed :) What made you think of this specific idea? Did it just happen, or you got inspired by something? A book perhaps?
I’m not sure! I definitely learnt this condition that two vectors can’t be mapped to the same thing in class. Calling that ‘not losing info’ probably was just because I really like info theory so that’s what makes sense to me. I’m sure there are lots of other very clear ways to understand these things- it’s just personal preferences I guess!
Yes, it makes sense. Thanks for taking the time to reply :-)
For the case of square matrices, I like to think that since nxn matrices are geometrically n-dimensional oriented parallelopipeds, if their determinants are zero, i.e. their areas are zero, it won't make sense to take their inverses. Also, I like to think that geometrically, an inverse of a non-degenerate square matrix is simply that same parallelopiped, but oriented in the opposite direction.
This makes me wish school taught about functions and relations and whatnot before teaching linear algebra.
It does. Linear Algebra isn't taught until college; long after functions in high school prealgebra.
@@SlipperyTeeth for me I learned linear algebra in high school, and calculus didn't get much into what a function actually is, instead I didn't learn that until a 2nd year foundations of mathematics course
Yeah, it varies a lot around the world. For me, we learnt a lot about functions and their meaning early, but when we did Linear Algebra it was all just about linear equations in school. It was only in uni I realised matrices are just very special functions! Would have helped to have know right?
This is wonderful!
Oh thank you so much! I was really worried about it.
"Let B be a matrix that takes 3D vectors to 2D that does rotate this place back. It is a left inverse of A."
The question then asked out loud is:
"Is B A's left inverse" which is the same as "Is B a left inverse of A" but this was already stated. The poll question, however, is correct to instead of if A is B's left inverse.
I think it says, ‘Is A B’s left inverse?’ Which is correct, no?
I loved this very much. Its been a long time since I have through of it. For me a matrix is always square and i always need to find the inverse. The inverse always exist if i have formulated the question correctly.
Me too! My matrices are always unitary. So I had to relearn a bunch of stuff for this video :P
For the matrices A and B rotating the plane into 3D and back again, I'm a little confused why B couldn't be a Left inverse of A. In the counter-example, B(v) isn't just rotated into the 2d plane, it's also pushed somewhere random. But why would B do that? Doesn't B just rotate the things into the 2D plane? B still sends everything uniquely into the plane, right? Or is it that not everything in the 3D space gets sent to the 2D plane at all? Like for some B(v) (which just does the rotation) won't end up in the plane that A cares about?
Can a left inverse map one vector to multiple values and still exist?
I never quite understood the purpose of using the determinant to check if a matrix has an inverse. It's equally as hard to compute the determinant as to compute the inverse itself, so to me it would make more sense to try to compute the inverse and just check if that fails.
Brilliant!
👏👏👏
Can a matrix contain a set? Like can you do (read like it was vertical) {4 5} . {1 0} = {4 0} and make the inverse {1 N} with N being all natural numbers?
There’s nothing stopping you from scaling a basis vector by a set
While you’re at it, why not have a vector with other vectors as components, what about a matrix as a component?
You can have derivatives if matrices with respect to vectors scaled by complex numbers of you really wanted to
What is a clifford algebra
Shouldn't q2 of the video at 6:12 say does L(|v》) exists ALWAYS? If b=0 then it does exist.
It’s been a long time!
Sorry! Thanks for waiting!
U have given a link of vector space brilliant course can u regive it because I can't buy a brilliant subscribtion right now
I’m really sorry, they only made that part free for 2 weeks (which are over now).
too good.
awesome!
Thank you!!
Ur vids jus beautiful 💙
Can’t a Hilbert curve map points from a 1D line to 2D space and the reverse without losing information. Perhaps the fact that it is bounded means that it doesn’t contradict your video, or is it because it is a non-linear transformation? Anyhow thanks for the great video!
Excellent question! I was thinking about this a lot. So, you can (and should!) prove that left invertible matrices ‘preserve dimension’ (the image space is the same dim as the original space). The reason for this is the linearity. So because of this, if the image space isn’t the whole target space, the matrix isn’t left jnvertible. But! As you pointed out, Hilbert curves are not linear and are invertible- which is pretty cool!
Hey hi you're back!
Hi, thanks for coming back!
@@LookingGlassUniverse lol i missed this channel
I multiply four times the first row with first column. Only the last gives me 1 so the answer is 4
The matrix in the video 1 0
0. 0
Has an inverse because it's deterpmininent is not 0
Thanks again LGU - you don’t need to make everything dramatic for us to enjoy it. WHY is the TRACE important? In Nima Armani-Hamed’s video’s, he is always saying, “You trace over this and you trace over that.”
Do you mean the partial trace in QM? It’s because when you say you trace over a system, you mean that you throw that system out, and look at what the remaining stuff looks like.
I am trying to find a video to teach my child how to create these kind of whiteboard presentation video on RUclips your guide n help would be much appreciated if you can share this knowledge so I can teach my children
One easy way to do it is to download a stop motion app on your phone that takes 1 photo a second and stitches it together as a video. You’d then need to slow it down. You’ll need some sort of stand (eg, on that clips onto the table) to hold your phone above the board. Hope that helps!
Are there right inverses?
Yes :) if you have AB=identity then B is a right inverse of A. The reason I think Left inverses make more sense is because you do the thing on the right first then you undo it with the left inverse.
Looking Glass Universe Figured it’d be that, thanks :-)
You have a pretty voice.
And last but not least :
Thank you for this video. Very eye opening!
Thank you!
13:17
Pretty good. Now imagine an axis perpendicukar to each vector. (Spoiler: its easy, think of time)
Now imagine a number that does that. (It has perfect name, the imaginary number)
So now just think of complex linear algebra as chucking in phase everywhere.
You could zip through the last 3 videos again and add lil clocks to your vectors, bumping them up a dimension. Should be informative.
Now just soak in the horrible truth that time is a 4th imaginary dimension, and bam! Quantum mechanics done.
Now, every particle has 3 familes, so we might as well redo this trivial process with quaternions. What is the result?
Got to say, I'm disappointed to see you doing linear algebra, given that you do QM better than anyone online. But I do also enjoy your take on linear algebra. I don't really need to know and I'm not all that interested, but it would have been really nice to have this video back in the fall of '99 when I took Calc 3 at Cornell as my very first college class and was surprised to see my teacher with a bearly intelligible accent draw this thing on the board called a matrix. And I didn't really know why I needed to know until I took QM 3 semesters later, by which time I had forgotten all of it.
Thank you :) I know linear algebra is not my strongest suite, but I really like making them. Plus, the QM videos are so important to me, I want to get them right- which means I end up getting very perfectionistic about them. This is nice because it lets me keep making things and getting into the rhythm of making regular videos and this new style so that when I get back to QM it’s easier.
next zero divisors! :D
That could be interesting! But I’m sorry- I have a hundred planned videos at this point :’(
baseisis
I win
Want to understand anything in maths, learn to draw it first.
A foot!
Where?? :P