Brilliant explanation! I have only one complaint. I have read (though, not thoroughly understood) both of your linear algebra books and your point in the beginning of this video of "flipping your thinking", pointing out that x and y are constants in "linear world" would have been quite helpful in my early attempts at understanding linear algebra. My last formal math class was about 50 years ago so x,y and a,b as variables and constants are pretty much hard coded in my brain, so I need all the help I can get. In all seriousness, great books and great videos!
I think that this was a great explanation for quadratic equations, but it’s not generalized if you don’t know the order of the polynomial, right? We can’t make the assumption that we have enough points to solve unless it’s explicitly stated in the question (in which cases we have at least enough and reducing solves it)- but we normally can’t in the real world.
It actually works even if you don't know the degree of the polynomial. If you're given n points, you can just guess that it'll be degree n-1 then solve the linear system. The worst thing that might happen is that some of the leading coefficients might equal 0 in your solution, so the polynomial is lower degree than you expected. For example, if you're given 3 points you should always guess "parabola" like we did in the video, but sometimes after you solve the linear system you might get a = 0, so the "parabola" is just y = bx+c, which is a line.
Brilliant explanation! I have only one complaint. I have read (though, not thoroughly understood) both of your linear algebra books and your point in the beginning of this video of "flipping your thinking", pointing out that x and y are constants in "linear world" would have been quite helpful in my early attempts at understanding linear algebra. My last formal math class was about 50 years ago so x,y and a,b as variables and constants are pretty much hard coded in my brain, so I need all the help I can get.
In all seriousness, great books and great videos!
Thanks! And yeah, that's a very good point, and a very common "mental roadblock" for people.
I think that this was a great explanation for quadratic equations, but it’s not generalized if you don’t know the order of the polynomial, right? We can’t make the assumption that we have enough points to solve unless it’s explicitly stated in the question (in which cases we have at least enough and reducing solves it)- but we normally can’t in the real world.
It actually works even if you don't know the degree of the polynomial. If you're given n points, you can just guess that it'll be degree n-1 then solve the linear system. The worst thing that might happen is that some of the leading coefficients might equal 0 in your solution, so the polynomial is lower degree than you expected. For example, if you're given 3 points you should always guess "parabola" like we did in the video, but sometimes after you solve the linear system you might get a = 0, so the "parabola" is just y = bx+c, which is a line.
Hey king, have you thought about covering Measure Theory in a similar manner to how you covered Advanced Linear Algebra 🫣
It's on my seemingly endless list of videos and lecture series that I'd like to do :)