How do I do this?

How do I do this?

by not being a brainlet

The third order derivative?

Its been years since I've done this. Its not hard though.

Can you explain it to me?

What's [math]W

Nope, can't remember how to do it.

What is F(-inf,inf) and is P3 the set of polynomials of degree 3 right?

F(-inf, inf) is all continuous functions, P3 is degree three polynomials.
W & V are vector spaces.

I am sure you meant the 4th order derivative
This doesn't work because then the kernel would include P2, P1 AND P0 aswell.

Nigga what are you smoking?
Look. If we take P3 to be the set of polynomials of degree 3 it's trivial to see there are no such linear maps, because P3 is not a group under addition. Think about it, T(x^3+x)=0 and T(x^3)=0 but T(x^3+x-x^3) != 0? Can't happen.
If we take P3 to be the set of polynomials of degree at most 3 then the fourth derivative is good.
/thread

[math]P_3[/math] includes those spaces. Otherwise, it wouldn't be a vector space (zero wouldn't be in there).

Ah fuck you are right.

OP confused me and made me think this was a non-trivial problem. I mean, the fourth derivative is clearly the fucking answer. But if strictly the 3rd degree polynomials had to be the kernel then that would be non-trivial.

OP you fucking retard kys.

Can you explain why?

Not OP, but this thread got me interested so I'm looking at other problems in that section of the book

My guess for the answer to this one is all strictly odd polynomials, is that right?

Wait wait wait wait wait WAAAAAAAAAAAAAAIT.

I just started thinking really hard.

If V is the vector space of real-valued functions with continuous derivatives... then wouldn't the fourth derivative map some piecewise functions to 0?

Like, take a third degree polynomial and a two degree polynomial. You can find some of those that fit together at some x so that they keep their differentiability.

You motherfuckers. This problem really is non-trivial.

I am sorry OP. We live in a world brainlets.

I got this one.

Ker(J)={p(x)=a+bx | a=0, bER}

no just normalize the vectors and do a lagrangian on the resultant identity matrix

V is the space of real-valued functions with continuous derivatives of All Order. I might be naive, but go ahead and show me a piecewise analytic function

Holy fucking shit nigga just kill yourself
Take the function f defined everywhere on R to be 0
Integrate 4 fucking times
You now have the set of all functions that have a fourth derivative everywhere equal to 0
This set is called the fucking set of polynomials of degree at most 3
This is fucking retarded
It's absolutely obvious
The integral of ax+b from -1 to 1 is 2b
Thus b=0 and a can be any real number

>My guess for the answer to this one is all strictly odd polynomials, is that right?

No.

J only maps polynomials of degree 1.

Under this transformation, the kernel would be polynomials of degree one that are symmetric in the interval (-1,1) with respect to x=0

So you want functions so that f(-x) =-f(x) around (1,-1) so you fix

-bx + c = -(bx + c)
iff
-bx + c = -bx - c
iff
c = -c
iff
c=0

>piecewise functions

This is where you fail

That is an odd function. The entire set of odd functions map to zero, not just lines

It's not just lines that map to zero, it's all odd functions, just look. The very definition of an odd function is one that is symmetric about the origin

J is defined on the space of linear maps you fucking mongoloids.
How can its kernel contain objects not in its domain of definition?

>MFW I read this thread
You are all FUCKING BRAINLETS

Oh, just noticed the input space is P1 anyway. So whatever, both answers are technically right

But the problem clearly says P1.

If it were the broader context of all functions then to find all answers you would have to get into trig functions and other shit aswell. Impossible to parametrize.

Read the problem correctly, the domain is P_1

what's a "domain"?

You were useful, everyone else was not. Thank you. Thread will be deleted

He's wrong retard

I can visualize the answer so it's good enough for me.

>I can visualize the answer
Fucking hippies I swear

noting that V is contained in W, choose an inner product on V (possible due to the axiom of choice), and let T be orthogonal projection onto the orthogonal complement of P_3.

Ok guys talked to the professor, it doesnt have a solution!

What about the fucking 4th derivative you cuck

HAHAHA NIGGERS JUST KILL YOURSELVES
FUCKING BRAINLETS I SWEAR
Veeky Forums is just a circlejerk that does nothing but pretend they are smart when they are a bunch of fucking inbred mongoloids
If you can't do this shit you honestly have no place here
>or anywhere else desu
I've seen monkeys smarter than you. Honestly, leave this board and go read a book or a hundred. I suggest you start with kindergarten coloring books, those might be appropiate for you. Work your way up until you understand this babby shit-tier year 1 stuff or kys

I can't believe you fell for my bait