Not that it does't exist but that it is not unique, ... in fact it is a subspace, ... infinitely many solutions, depending on a parameter we usually call "constant"
7:50 I'm guessing that matrix works for any polynomial of degree three or lower since it's composed of the would-be exponents of the terms within such a single variable polynomial.
yeah for every polynomial of degree n there is this matrix T which is (n-1) times n. Where that diagonal from element (1,2) contains the positive integers up to n. But indeed you extend it with the idea of the would-be components, and by placing every entry of the input vector one place down in the output vector. (bc it's n-1xn)
Is P_3 really the notation of the space of the polynomials with degree less than or equal to 3? Weird notation, never saw it before, I know the notation K_3[X], where K is the field of the coefficients of the polynomials. In particular for polynomials with coefficients in R and with degree less than or equal to a natural number n, we have R_n[X].
It's literally the only notation I've ever seen. The notation you suggest isn't something I've seen. It doesn't play well with field notation for finite fields.
Yes Crazy E, but I find K_n[X] better, because it gives 3 information: 1. The polynomial is with degree n. 2. His coefficients are in the field K. 3. The indeterminate is X. To clarify what I'm saying, R_n[X] and C_m[Y] are different in every single way, the fields are different, the degrees are different (assuming n=/=m), and the indeterminates are different.
That seems like a crazily inefficient way to differentiate in software. Lots of pointless multiplications by zero. If a_0, a_1, a_2, a_3, ... are the coefficients of the input, and b_0, b_1, b_2, ... are the coefficients of the derivative, then why not just do: b_0 = a_1, b_1 = 2 * a_2, b_2 = 3 * a_3, ... b_(n-1) = n * a_n ?
And of course the matrix is not invertible...
Which proves that integration doesn't exist.
QED
Not that it does't exist but that it is not unique, ... in fact it is a subspace, ... infinitely many solutions, depending on a parameter we usually call "constant"
1:43 I believe I can fly, I believe I can touch the sky
7:50 I'm guessing that matrix works for any polynomial of degree three or lower since it's composed of the would-be exponents of the terms within such a single variable polynomial.
It works for any polynomial, you would just have a different sized matrix
yeah for every polynomial of degree n there is this matrix T which is (n-1) times n. Where that diagonal from element (1,2) contains the positive integers up to n.
But indeed you extend it with the idea of the would-be components, and by placing every entry of the input vector one place down in the output vector. (bc it's n-1xn)
@@skeletonrowdie1768 This is be out of context, but are you a Queen fan?
Why do you mark your basis vectors with exponents, while they are typically covariant?
At this level it doesn’t matter, and I’m not a geometer, so that’s that, lol
Is P_3 really the notation of the space of the polynomials with degree less than or equal to 3? Weird notation, never saw it before, I know the notation K_3[X], where K is the field of the coefficients of the polynomials.
In particular for polynomials with coefficients in R and with degree less than or equal to a natural number n, we have R_n[X].
It's literally the only notation I've ever seen. The notation you suggest isn't something I've seen. It doesn't play well with field notation for finite fields.
Yeah, P3 is legit
In the end it is all just notation
Yes Crazy E, but I find K_n[X] better, because it gives 3 information:
1. The polynomial is with degree n.
2. His coefficients are in the field K.
3. The indeterminate is X.
To clarify what I'm saying, R_n[X] and C_m[Y] are different in every single way, the fields are different, the degrees are different (assuming n=/=m), and the indeterminates are different.
@@patricksalhany8787 alright, it's the superior one. It is the VIP in the party of notations. But seriously, I am on your side
Thank you!
A very cute problem, he said 💖🎀😊🎀💝
Dr. Peyam after dark
That seems like a crazily inefficient way to differentiate in software. Lots of pointless multiplications by zero.
If a_0, a_1, a_2, a_3, ... are the coefficients of the input, and b_0, b_1, b_2, ... are the coefficients of the derivative, then why not just do:
b_0 = a_1,
b_1 = 2 * a_2,
b_2 = 3 * a_3,
...
b_(n-1) = n * a_n
?