I just realized something. A function that is said to be linear has the property of having a constant derivative. But functions can also be seen as transformations, something that takes a bunch of points in a space and takes them to another (possibly the same) space. And the Derivative in this context has the interpretation of being the factor by which the space gets stretched of squished during that transformation. So, when the function (transformation) is linear, the constant derivative means that space is being scaled by a constant amount, and that is tantamount to saying that grid lines remain paralel and evenly spaced, which is the geometrical interpretation of linearity in 2D in the linear algebra context. Sorry if sounds messy (or even wrong haha). Just sharing a thought.
So if i want to expand to a higher order, do i just use somthing very similar to a taylor expansion, or does it get more complicated than that in more than 1D?
How do you measure the error? I meant, it is just an approximation, so when should I know that the error has exceeded certain tolerance/threshold and thus I should compute a new linearization closer point?
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I just realized something. A function that is said to be linear has the property of having a constant derivative. But functions can also be seen as transformations, something that takes a bunch of points in a space and takes them to another (possibly the same) space. And the Derivative in this context has the interpretation of being the factor by which the space gets stretched of squished during that transformation. So, when the function (transformation) is linear, the constant derivative means that space is being scaled by a constant amount, and that is tantamount to saying that grid lines remain paralel and evenly spaced, which is the geometrical interpretation of linearity in 2D in the linear algebra context. Sorry if sounds messy (or even wrong haha). Just sharing a thought.
Next time, keep your thoughts to yourself.
@@tiny_toilet shut up .
@@darkseid856 no u .
@@tiny_toilet f u
@@elibaum5489 l l h o u s e
So if i want to expand to a higher order, do i just use somthing very similar to a taylor expansion, or does it get more complicated than that in more than 1D?
Hold tight! The next few videos will be on Quadratic approximations of multivariable functions :)
@@khanacademy but could you please till where is the article that you mentioned during the Video
thank you
So, this linearization is the directional derivative, where the vector i dot with the gradient is the Vector=[ dx , dy ,...] ?
So where can I find the post about linearization and practice it ? ( mentioned at about 9:03 )
Same question 3 years later
@@chengzhiwu7052 u can check it at khan academy wibsite
same quetion 3 years, 7 months later
How do you measure the error? I meant, it is just an approximation, so when should I know that the error has exceeded certain tolerance/threshold and thus I should compute a new linearization closer point?
Anyone know of the article he mentioned?
looks like taylor-approximation
there is the article please
Locally near
Local linear ;O
Does anyone know how to find this article he's talking about?
Most likely it's on Khan Academy
where's sal
This guy is better than Sal.. he uses animations to clearly explain stuff, rather than go on and on in theoretical manner with tons of equations..
I'm also missing him. The voice of this guy makes me sick; and talk too much.
dude. this is like the 75th video with Grant Sanderson.
GRANT>>sal
Shall we just agree that both guys are great teachers?