One-to-one, Onto, and the Big Theorem Part II

Learning Objectives:
1) Define one-to-one and onto
2) Given a matrix, decide whether it is one-to-one or onto
3) Find equivalent properties to a transformation being onto (or one-to-one)
This video is part of a Linear Algebra course taught at the University of Cincinnati.
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[@mostafaahmadi4998]

The first 30 simple, clear, and concise videos - each on average 10 minutes, adding up to approximately 300 minutes ~ 5 hours - bring together seemingly different ideas in linear algebra in such a way that NO BOOK ON THIS PLANET CAN DO THE SAME JOB by reading that book for five hours. Every single video of yours deserve to be liked. Thank you for all the time and effort you have put in creating the Linear Algebra playlist. I really hope that all of you other playlists have this same incredible effect in them.

[@rounakgour6370]

I think you well expressed what each of us viewer feel after watching these amazing videos that are just to the point and well explained

[@NguyenHaNhutLong]

No one has ever explained visually about 1-1 and onto like you did! Before I watched your video, I just knew that 1-1 that A is linear independent, but now I understand why. You did a great job! Thank you so much!

[@mayanknaithani5412]

Thanks a lot, best Linear algebra course on internet

[@Barba_007]

@2:12 what happened there lol xD

[@paschikshehu7988]

I'm dying

[@Bruhhhhhhhhhhhhhhhhhhhhhh]

Nam flashbacks

[@austinnguyen9107]

@@Bruhhhhhhhhhhhhhhhhhhhhhh I felt that

[@austinnguyen9107]

Glitch in the matrix

[@naiko1744]

Jokes aside, it's way cooler to take the time to rephrase rather than to explain bullshit

[@syedmasroorhussain]

I love this dude so much

[@IzzyP688]

Thanks for the help man. Passed my quiz because of you.

[@youssefdirani]

I think one-to-one is called injection in french and onto is called surjection.

[@johnhippisley9106]

Right, and if it is both one-to-one and into it is a bijection

[@marcusaurelius2147]

Right, and and if the transformation is bijective, it is an isomorphism

[@xoppa09]

Condition 4 is equivalent to there being no zero rows for REF A or RREF A .
I assume that it does not matter if REF A or RREF A is used here.
So R(R)EF A having no zero rows implies Ax = b is onto , for all b.
Since if there was a zero row we have the inconsistent augmented matrix
* * * * * *
[0 0 0 ... 1] , thus not all b's are mapped to by T(x), namely all the b vectors which have a nonzero element in its final component.
The one-to-one type Ax =b would probably occur when every column in A is a pivot column, i.e. there are no 'free variable' columns.

[@xoppa09]

yes i am watching your videos over again at speed to let the 'big ideas' sink in. very nice

[@xoppa09]

9:17 Is it possible to check my equivalent big theorem with one-one.
I don't know if all the conditions have analogous parts.
Here is my attempt:
TFAE
1) The rows of A span R^n
2) not sure
3) not sure
4) All the columns of A have leading ones.
5) The transformation Ax is one-one

[@cesarmoreno1675]

you are a gift!

[@lexixu3655]

thank you

[@MangoLassiYT]

is onto also called range of T

[@kamrulhassan7157]

at 2:12 I do not understand.

[@continnum_radhe-radhe]

🔥🔥🔥