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## Homework Statement

Let ##n>1\in\, \mathbb{N}##. A map ##A:\mathbb{R}_{n}[x]\to\mathbb{R}_{n}[x]## is given with the rule ##(Ap)(x)=(x^n+1)p(1)+p^{'''}(x)##

a)Proof that this map is linear

b)Find some basis of the kernel

b)Find the dimension of the image

## Homework Equations

##\mathbb{R}_{n}[x]##

is defined as the set of all polynomial with real coeficient that have the power less or equal to n

##kerA=\{x;Ax=0\}##

##imA=\{Ax,x\in\mathbb{R}_{n}[x]\}##

##p(x)=a_{n}x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\ldots+a_{2}x^2+a_{1}x^1+a_{0}x^0##

## The Attempt at a Solution

a)Prove that this map is linear

##p,q\in A \\

(A(p+q))(x)=(x^n+1)(p+q)(1)+(p+q)^{'''}(x)=((x^n+1)(p)(1)+(p)^{'''}(x)+(x^n+1)(q)(1)+(q)^{'''}(x)=Ap(x)+Aq(x)

\\

\text{ }

\\

A(\theta p)(x)=\theta((x^n+1)(p)(1)+(p)^{'''}(x))=\theta(Ap)(x)

##

c)

I started doing this by writing out the possible polynomial

##(Ap)(x)=(x^n+1)p(1)+p^{'''}(x)\\

p(1)=a_{n}+a_{n-1}+a_{n-2}+a_{n-3}+\ldots+a_{2}+a_{1}+a_{0}\\

p^{'''}(x)=6a_{n}\binom{n}{3}x^{n-3}+6a_{n-1}\binom{n-1}{3}x^{n-4}+\ldots+6a_{4}\binom{4}{3}x+6a_{3}

##

now I placed all of this together

##

(Ap)(x)=(x^n+1)\displaystyle\sum_{i=0}^{n}a_{i}+6a_{n}\binom{n}{3}x^{n-3}+6a_{n-1}\binom{n-1}{3}x^{n-4}+\ldots+6a_{4}\binom{4}{3}x+6a_{3}

##

Then I paired all of the same coeficients together and got

##

a_{n}(x^{n}+1+6\binom{n}{3}*x^{n-3}) \\

a_{n-1}(x^{n}+1+6\binom{n-1}{3}*x^{n-4}) \\

a_{n-2}(x^{n}+1+6\binom{n-2}{3}*x^{n-5}) \\

\vdots\\

a_{3}(x^n+7) \\

a_{2}(x^n+1) \\

a_{1}(x^n+1) \\

a_{0}(x^n+1) \\

##

Here I noticed that the bottom 3 functions are linearly dependent, which means that If I want to find the basis or dimension I should take ##a_{0}\, and\, a_{1}## out. Then I also noticed that all of the above (a

_{n}to a

_{3}) are also linearly dependent on a

_{2}so I subtracted them and got

##

a_{n}(x^{n-3}) \\

a_{n-1}(x^{n-4}) \\

a_{n-2}(x^{n-5}) \\

\vdots\\

a_{3}(6) \\

a_{2}(x^n+1) \\

##

as my basis for the image of A therefore the ##dimension(imA)=n-1##

This is as far as I have gotten. I've tried solving b) by setting the whole polynomial I got equal to 0 however I have no idea how to continue from there.

thank you

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