Construction of a linear transformation with specific kernel and image

Question

I got stuck on this exercise and its really frustrating :/

It says something like this:

Given W={p(x) ∈ P₂[R] : P(1)=P(-1)} a linear transformation f:P₂[R] --> P₂[R] is required such that Ker(f) will be W

1)What should i know about W to know if i can get any linear transformation with the previous information?

2)Is it possible to get the linear transformation such that the polynomial h[x]= X will belong to Im(f)?

I dont understand the exercise.. take a look:

For 1) i think i should know the basis of W because its a linear transformation

I did this:

P₂[R]=ax^2 + bx + c

P(1)-P(-1)=0

a + b + c - (a - b + c)=0

a + b + c - a + b - c = 0

2b = 0

b=0

Then,

P₂[R]=ax^2 + c P₂[R]=a(x^2) + c(1)

Basis=(x^2,1)

What should i do now? I have no idea what to do :(...

What about point 2)?

Thanks!!

Answer 1

You just have to define an appropriate application on the basis vectors. For instance, you want $W$ to be the kernel, right? Then take the basis $\{x^2,x,1 \}$ and send $x^2$ and $1$ to $0$ and $x$ in $x$. This way the kernel is $W$ and $Span(x)$ is the image.