Invertible Matrices correspond with Invertible Transformations **proof**

Invertible Matrices are an algebraic concept that helps us solve Linear Systems of Equations. Invertible Transformations are a geometric concept where we can "undo" a transformation. But in fact they coincide! In this video, we prove that if you have an invertible matrix, the transformation it defines is also invertible. Likewise, if you have an invertible transformation, the matrix it generates is invertible.

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Now it's your turn:
1) Summarize the big idea of this video in your own words
2) Write down anything you are unsure about to think about later
3) What questions for the future do you have? Where are we going with this content?
4) Can you come up with your own sample test problem on this material? Solve it!

Learning mathematics is best done by actually DOING mathematics. A video like this can only ever be a starting point. I might show you the basic ideas, definitions, formulas, and examples, but to truly master math means that you have to spend time - a lot of time! - sitting down and trying problems yourself, asking questions, and thinking about mathematics. So before you go on to the next video, pause and go THINK.

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This video was created by Dr. Trefor Bazett, an Assistant Professor, Educator at the University of Cincinnati.

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[mostafaahmadi]

The side by side comparsion with the inclusion of the transformation animation was just incredible. Thank you very much.

[weisanpang]

Dr Trefor, in this video you talked specifically about a matrix being Onto in order for it to be invertible, however, in a previous video you also specifically said that an invertible matrix must be one-to-one, which is a distinct property of matrix from being Onto. The fact that an invertible matrix must be reducible to rref form with all columns being a pivot column means that the matrix is one-to-one. Correct me if I'm wrong, but I think Onto itself doesn't make a matrix invertible be definition. However, it also seems for any n-n square matrix, if it's onto, it must be one-to-one as well.

[animeshbasakchowdhury]

@Dr. Trefor Bazett. Thanks for the videos. I have a question when you derive the proof for Invertible transformations leads to invertible matrices.

[maurocruz]

Amazing video! I have a question.
What about the transformations which only have right or left inverse. Is there a way to prove they have an associated mxn (non squared) matrix (i.e, a lateral inverse)?

[roshanghoti]

Sar Class 11th 12th maths ke video banay pkzzz replay kejeye