I don't know if you wanted this explanation but if you do: linear algebra deals with fully linear transformations, which are just the small subset of these vector to vector functions that look like f(xi + yj) = (ax + by)i + (cx + dy)j where a,b,c,d are all just constants. These transformations keep all the gridlines straight and evenly spaced globally, throughout the space, not just when you zoom in on a point
Counter-example: |x| is continuous at x=0, but not differentiable at that point. +elbay2, A better condition for what you described is that the function must have well-defined partial derivatives at that point (which |x| has everywhere except x=0). This is kind of my first time really seeing this locally linear stuff, so I don't really have any examples in mind; just the concept of differentiability. A hint for this viewpoint is found in the Jacobian matrix, which is built up of partials. I hope this helps. Sorry if I got anything wrong #NewLearner. -Float Circuit.
@@ozzyfromspace if you were to play the animation of the transformation of |x| every positive input would map to itself which means this function is locally linear at all regions that don't include negative values
@@lukemoura256 actually it would be locally linear at every point except for zero. That's because the condition for local linearity is differentiability. And a sufficient condition for differentiability is the existence of continuous partial derivatives.
3blue1brown's animations are amazing
Indeed!
At my university, we have to take multi variable calc before linear algebra so I'm sort of unfamiliar with this explanation
I don't know if you wanted this explanation but if you do: linear algebra deals with fully linear transformations, which are just the small subset of these vector to vector functions that look like f(xi + yj) = (ax + by)i + (cx + dy)j where a,b,c,d are all just constants. These transformations keep all the gridlines straight and evenly spaced globally, throughout the space, not just when you zoom in on a point
such amazing content
Thank You Sir!
Could you pleeeeeease metion the name of the program you used for this transformations ??
Question: are all functions that are made up of various combinations of elementary functions locally linear, or are there some that are not?
It looks to me that so long is a function is continuous it will be locally linear, and I can't think of a counterexample off the top of my head.
Thanks!
Counter-example: |x| is continuous at x=0, but not differentiable at that point.
+elbay2, A better condition for what you described is that the function must have well-defined partial derivatives at that point (which |x| has everywhere except x=0). This is kind of my first time really seeing this locally linear stuff, so I don't really have any examples in mind; just the concept of differentiability. A hint for this viewpoint is found in the Jacobian matrix, which is built up of partials.
I hope this helps. Sorry if I got anything wrong #NewLearner.
-Float Circuit.
@@ozzyfromspace if you were to play the animation of the transformation of |x| every positive input would map to itself which means this function is locally linear at all regions that don't include negative values
@@lukemoura256 actually it would be locally linear at every point except for zero. That's because the condition for local linearity is differentiability. And a sufficient condition for differentiability is the existence of continuous partial derivatives.
This should have more views.
Just awesome, thanks!
hail 3 blue 1 brown!!
thank you so much
Can you please share the program you use for the transformations?
Fernando Franco Félix www.3blue1brown.com/about/
Its called manim
The hero we need but don't deserve
Is there a way to find out what playlist this video is part of? I want to start from the beginning... at least previous video?
th-cam.com/play/PLSQl0a2vh4HC5feHa6Rc5c0wbRTx56nF7.html
WOW
Thank you!
amazing!!!
what program is he using to show all that?