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

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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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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.

mostafaahmadi

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!

NguyenHaNhutLong

Thanks a lot, best Linear algebra course on internet

mayanknaithani

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

IzzyP

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

youssefdirani

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.

xoppa

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