Finding the matrix of a linear transformation and figuring out if it's one-to-one and/or onto. Check out my linear equations playlist: • Linear Equations
For onto, I think about the space of all inputs (x) as an island and the space of all outputs as another island. I imagine the transformation A to take anyone in (x) to a place in the outputs (b). And onto means that there will always be an inverse bridge that takes anyone in (b) to a place in (x).
Yay! Great video as always! The answer to those 2 questions is very dependent on the space considered. If your image space is {(0,x,y,z) l (x,y,z) ∈ lR^3}, then the results differ, don't they?
Why do English speakers always use "1 to 1" and "onto"? Here we always say injective and surjective (and bijective instead of "1 to 1 and onto" for that matter). Just something I always wondered.
Beyond linear algebra it's pretty much never used. Though I can defend 1 to 1 because it's often useful to talk of " to ". Though the phrase "maps onto" instead of "maps surjectively" is still used.
Hello! Thank you so much for your video, that was really helpful! But I have one question when mentioning the pivot positions in the rows and columns, do we consider only the coefficient matrix or the augmented matrix?
Would this be a valid and simpler argument without considering the associated matrix A at all? Every output of T has 0 in the first coordinate so it's clearly not onto. Furthermore, it's range is at most 3 dimensions so it must be many to one.
Hi Dr. Peyam! I haven't seen all your linear algebra videos; can you tell me if you already made a proof for the Rouché-Capelli/Kronecker-Capelli/Rouché-Fontené/Rouché-Fröbenius theorem? (All the names are different ways for naming the same theorem). If not, can you make a proof? I really need it to understand my classes. Thank you very much if you read this.
@@drpeyam It's funny because in Spain, where I live, is like a fundamental theorem, however, I always prefer to search math content in English but I barely see videos about that theorem. I would be so pleased if you dedicate one video for it :)
@@drpeyam It is a great theorem for discussing linear systems of equations with some parameters. Let A be the coefficient matrix and A* be the augmented matrix of a system of linear equations; let n be the number of variables the system has. Then the theorem states that if rank(A)=rank(A*)=n, the system has exactly one solution; if rank(A)=rank(A*)
I remember the theorem. For a linear map from a vector to itself: injective if and only if surjective. It's a consequence of the dimension. Of course, that's not really good to bring up yet, pedagogically.
That's the terminology I learned in first course Calculus and Linear Algebra and the one I mainly use. One-to-One is also ok, but 'onto' hasn't even got a proper translation in my daily language. However as long as English is concerned, I think all these words are equally worthy. Probably 1-1 and onto are even more suitable for a first course
Just wanted to say thank you for all the linear algebra videos, Dr. Peyam!
Thanks so much!!!
Instablaster...
my kind sir, I don't know why this was as hard to find as it was as it is a simple concept. You explained it beautifully, thank you.
For onto, I think about the space of all inputs (x) as an island and the space of all outputs as another island. I imagine the transformation A to take anyone in (x) to a place in the outputs (b). And onto means that there will always be an inverse bridge that takes anyone in (b) to a place in (x).
Bro thank you so much, been struggling to figure this out for my exam, I kept getting confused with my teacher's terminology. THANK YOU!!!
Thank god you exist, the way most people teach this is so obfuscated 😭
Thank you!!!
You are a great teacher sir ❤️❤️
Thanks a ton!!
Yay! Great video as always!
The answer to those 2 questions is very dependent on the space considered. If your image space is {(0,x,y,z) l (x,y,z) ∈ lR^3}, then the results differ, don't they?
Definitely!
Why do English speakers always use "1 to 1" and "onto"? Here we always say injective and surjective (and bijective instead of "1 to 1 and onto" for that matter). Just something I always wondered.
That's proper mathematical English too, but we English speakers are sometimes lazy :)
Beyond linear algebra it's pretty much never used. Though I can defend 1 to 1 because it's often useful to talk of " to ". Though the phrase "maps onto" instead of "maps surjectively" is still used.
Thank you! Awesome video
You are just amazing!Thanks a lot!
You are a treasure, Sir!
Just read my textbook on linear transformations and your video perfectly complemented that.
Well explained, thanks.
Hello! Thank you so much for your video, that was really helpful! But I have one question when mentioning the pivot positions in the rows and columns, do we consider only the coefficient matrix or the augmented matrix?
Usually the coefficient matrix
30 minutes away from the online exam, thanks apr 26th 2021 11:00am
Can you please prove those alternative definitions of one to one and onto that you talked about? I'm intrigued!!
T is 1-1 if x not equal to y implies T(x) not equal to T(y)
T is onto B if for every b in B there is x with b = T(x)
Would this be a valid and simpler argument without considering the associated matrix A at all? Every output of T has 0 in the first coordinate so it's clearly not onto. Furthermore, it's range is at most 3 dimensions so it must be many to one.
Of course
Thanks alot doc
Hi Dr. Peyam! I haven't seen all your linear algebra videos; can you tell me if you already made a proof for the Rouché-Capelli/Kronecker-Capelli/Rouché-Fontené/Rouché-Fröbenius theorem? (All the names are different ways for naming the same theorem). If not, can you make a proof? I really need it to understand my classes. Thank you very much if you read this.
I’ve never heard of those theorems 😱
@@drpeyam It's funny because in Spain, where I live, is like a fundamental theorem, however, I always prefer to search math content in English but I barely see videos about that theorem. I would be so pleased if you dedicate one video for it :)
What does the theorem say?
@@drpeyam It is a great theorem for discussing linear systems of equations with some parameters. Let A be the coefficient matrix and A* be the augmented matrix of a system of linear equations; let n be the number of variables the system has. Then the theorem states that if rank(A)=rank(A*)=n, the system has exactly one solution; if rank(A)=rank(A*)
Oh, I didn’t know that had a name! Maybe in the spring I’ll talk about that
I remember the theorem. For a linear map from a vector to itself: injective if and only if surjective. It's a consequence of the dimension. Of course, that's not really good to bring up yet, pedagogically.
It’s a nice theorem, isn’t it?
Thanks sir ...
you are amazing
Hey Peyam!
Would you do a video about the Steinitz Theorem? ^_^
I don’t know what that is
@@drpeyam en.wikipedia.org/wiki/Steinitz_exchange_lemma
Oh, I didn’t know that had a name! Yeah, probably, but only once I teach the proofy linear algebra course, so in 3 months or so
@@drpeyam okay, sir ^_^
I am having a stroke watching this in x1.5 speed because the lights keep getting brighter and darker
Sorry
@@drpeyam don’t worry Dr peyam, the video was very informative! It was my fault for trying to cram on x1.5 and x2 speed, not yours
thank youuu
i love your watch
Thanks! :)
tyty
Nice! I'm watching this video in Budapest, btw :D still waiting for your arrow's spike to show up somewhere on the sky above me :-O
what jacket is that!!!!!
Bprp gave it to me
Would you mind, you all anglo-saxons people (you sure feel pointed out, Peyam 😂), using the right words someday? That means, injective and surjective
Seriously, why are you still using those hideous names?
That’s what they use in intro linear algebra classes
@@jonasdaverio9369 because they are beautiful
Jonas Daverio .....okay, but could you please say that in English?
That's the terminology I learned in first course Calculus and Linear Algebra and the one I mainly use. One-to-One is also ok, but 'onto' hasn't even got a proper translation in my daily language.
However as long as English is concerned, I think all these words are equally worthy. Probably 1-1 and onto are even more suitable for a first course