The determinant | Chapter 6, Essence of linear algebra

The determinant measures how much volumes change during a transformation.
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Hebrew: Omer Tuchfeld
Vietnamese: @ngvutuan2811

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Various social media stuffs:

[badlydrawnturtle]

That's what determinant is? Seriously? Why don't they just say that in the textbook? I spent days of my life wrestling with the idea that they wanted me to compute a magical number using an arbitrary formula.

[yujinpark]

This man deserves a Nobel Teaching Prize

[phillipliou]

These videos are truly amazing. Thank you so much for making them.

[spiralhalo]

As a programmer, knowing what these represent is astronomically more important than how to compute them for a good reason. Thank you.

[treyforest]

6:13 How can you say “parallelepiped” is the best shape name ever when mere seconds ago you were using the far superior “slanty-slanty cube”?

[slowpoke]

5 years after my first lecture on determinants I finally understand it’s nature

[lukitas]

I have a master's degree in mechanical engineering and I'm starting to think I should redo my whole education from ground up searching for this kind of intuitive knowledge. It's absurd that I find out explanations which are as intuitive as this one so late in my life. I'm blown away completely! I mean how many bits of information have we stumbled upon during our formal education failing to see how they elegantly relate to each other and form a bigger picture...oh my!

[alecyates]

"Understanding what it represents is, trust me, much more important than the computation"

Said none of my courses involving determinants over the past decade, and why years later I am still looking up this stuff on youtube! This channel is amazing.

[sasamilic]

You've achieved the impossible ... demystifying the determinant

[_strangelet__]

If the matrix M1 scales any area "A" to "cA", and M2 scales any area "A" to "dA", so this means that det(M1) = c and det(M2) = d, which implies det(M1)det(M2) = cd.

Now, if we consider the matrix M1M2, it is essentially like scaling the area "A" first by matrix M2, and then by matrix M1.
So, when we first transform "A" with M2, the area becomes "dA". Then, when we transform this new area "dA" with matrix M1, we know that M1 scales any area by a factor "c", so the new area becomes "cdA", hence we can conclude det(M1M2) = cd.

This shows that det(M1M2) = det(M1)det(M2).

[kavishkabartlett]

You have absolutely no idea how much your videos have made me appreciate linear algebra. I always understand the how and why, but never what everything actually represented. Don't have much to spare because I'm a broke college student lmao but your content is just helpful it wouldn't be fair to just take it for free. Hopefully everyone who watches donates at least a little so you can keep doing what you're doing!!

[UKXcalibur]

i almost started crying cause i finally understand what a determinant is. THANK YOU

[Piffsnow]

I'm a math teacher and I didn't even know all of that. Why nobody told us in uni ? These videos are great, really, but they would have been more useful to me 9 years ago. :/
Anyway, thanks a lot !

[helo]

I ask my teacher: What is the determinant?
My teacher: The determinant is the determinant!!!

[huh]

It might sound stupid but I nearly cried seeing this because for the first time since I started uni this year or even since I started middle school I feel like im deeply understanding the basic concepts and not just banging my head against the book trying ro get it in my head by memorizing, thank you from the buttom of my heart

[jacquesnicolay]

I'm a 4th year Mechanical Engineering student and have had algebra classes, calculus, trig, Engineering analysis, linear algebra etc... I just learned that a determinant is the factor by which a transformation is scaled lmao

[camkiranratna]

This is what I thought about the property Grant mentioned in the end.

Multiplying two matrices means that we are applying one transformation, then the other.

The first transformation scales a unit area by “c”, and the second transformation scales the scaled area by “d”. So the overall scaling for the 1x1 unit square is “c” times “d “ .

Now, looking at the right hand side we have the product of determinants. Since the determinants of the respective matrices are “c” and “d”, their product is “c times d”.

If anyone has a better explanation please let me know.

Thank you for your time .

[smortwonbon]

This video is so helpful, my uni never told us what a determinant actually is, we were just expected to compute it. This is really making me appreciate math a whole lot more and is motivating me to study harder. Thanks a lot!

[alejrandom]

Your videos have really changed my way of seeing mathematics. It's sad that the school system has grabbed something beautiful, cut off all the intuition, turned it into a chore, and just makes us memorize formulas instead of actually understanding the logic behind them. Math isn't about the numbers, it's about knowing how to go back and forth between the visual and the abstract. You sir, are my hero.

[MrRoflHamster]

Damn. That's actually interesting. At University the determinent was just that number. 'Here, go compute that, it's important, '