One to one, onto, matrix

Finding the matrix of a linear transformation and figuring out if it's one-to-one and/or onto.

[OneShotKill]

Just wanted to say thank you for all the linear algebra videos, Dr. Peyam!

[pepperbox]

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.

[itsguy]

Thank god you exist, the way most people teach this is so obfuscated 😭

[thedoublehelix]

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).

[anthonygracioppo]

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!!!

[TheTim]

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.

[hakbud]

30 minutes away from the online exam, thanks apr 26th 2021 11:00am

[benjaminbrat]

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?

[postnubilaphoebus]

You are a treasure, Sir!
Just read my textbook on linear transformations and your video perfectly complemented that.

[TheMauror]

Can you please prove those alternative definitions of one to one and onto that you talked about? I'm intrigued!!

[ldb]

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.

[sugarfrosted]

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.

[zhansayamaksut]

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?

[blooper]

I am having a stroke watching this in x1.5 speed because the lights keep getting brighter and darker

[DiegoMathemagician]

Hi Dr. Peyam! I haven't seen all your linear algebra videos; can you tell me if you already made a proof for the 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.

[SmileyHuN]

Hey Peyam!
Would you do a video about the Steinitz Theorem? ^_^

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

[jonasdaverio]

Would you mind, you all anglo-saxons people (you sure feel pointed out, Peyam 😂), using the right words someday? That means, injective and surjective