Math Help - Kernel and Range

1. Kernel and Range

Determine the kernel and range of the following linear transformation from P3 to R2:

L(p) =
( p'(0)
Integral from -1 to 1 of p(x) dx )

2. Originally Posted by harvardgurl29
Determine the kernel and range of the following linear transformation from P3 to R2:

L(p) =
( p'(0)
Integral from -1 to 1 of p(x) dx )
What in the name of God....

$L:\mathcal{P}_3\to\mathbb{R}^2$ given by $p(x)\mapsto \left(p'(0),\int_{-1}^{1}p(x)dx\right)$?

I'll help with the kernel. Let's see some effort on the image.

$\ker L=\left\{p(x):L(p(x))=(0,0)\right\}$

So, let $p(x)=a_0+a_1 x+a_2 x^2+a_3 x^3$.

We know that $p'(0)=a_1+2a_2\cdot 0+3a_3\cdot0^2=a_1=0$. Thus, $p(x)=a_0+a_2x^2+a_3 x^3$.

Furthermore, $0=\int_{-1}^{1}p(x)dx=a_0+\frac{a_2}{2}+\frac{a_3}{4}+a_0+\ frac{a_2}{2}-\frac{a_3}{4}=2a_0+a_2=0$

It follows that $\ker L=\left\{a_0+a_1 x+a_2 x^2+a_3 x^3:a_1=0,2a_0+a_2=0\right\}$