Linear transformation - linear matrix & kernel

Question

I have a problem understanding getting the KERNEL and IMAGE of a linear transformation. We have the following transformation given: $$ \mathbb{R}_{2}[ x ] \rightarrow \mathbb{R}_{2}[ x ] $$ $$ (\phi (p))(x) = (x p(x+1))' - 2p(x) $$

We first have to find its matrix in basis $$ \{ 1, x, x^2 \} $$ which I know how to get. The transformation matrix result is:

$$ \begin{bmatrix} -1& 1& 1\\ 0& 0& 4\\ 0& 0& 1 \end{bmatrix} $$

How do I get the KERNEL and the IMAGE from it ?

Would really appretiate an explanation, not just the result.

THANKS !

Answer 1

The image is generated by vectors that have the columns of the matrix as coordinates. Therefore the image is generated by $1$ and $1+4x+x^2$. These vectors are also linearly independent, so they form a basis.

The kernel is the set of vectors whose coordinates $X$ solve $MX=0$, where $M$ is the matrix.

Solving the system you get that the kernel is formed by vectors with coordinates $(a,a,0)$, thus a basis for the Kernel is the vector $1+x$.

Answer 2

To find the kernel just solve the homogeneous system

$$ AX=0,\quad X = [x_1,x_2,x_3]^T. $$

The image is the space spanned by the column vectors of $A$.