Determine a basis of $\operatorname{Ker} \ F$

Determine a basis of $\operatorname{Ker} F$ and one for $\operatorname{Im} F$, where $F:\Bbb R^4\to \Bbb R^3$ is the linear transformation defined by $$F(x_1,x_2,x_3,x_4):=(x_1+x_2+x_3+x_4, 2x_2+x_3+x_4,4x_2+2x_3+2x_4) .$$

I have no idea how to start. Any idea please? Thank you.

You could first find the matrix representation $A$ of $F$.
$A$ is the $3\times 4$ matrix whose $i^{\rm th}$ column is $F({\bf e}_i)$where ${\bf e}_i$ is the $i^{\rm th}$ unit vector in $\Bbb R^4$. You then have $$F({\bf x})=A{\bf x},$$ for all ${\bf x}\in \Bbb R^4$.
Then a basis for $\text{ ker}( F)$ is given by a basis for the null space of $A$ and a basis for the image of $F$ is given by a basis of the column space of $A$.
Let's start with the kernel. The kernel is the set of inputs yielding the output zero. So for us that's the set of solutions to the system, \eqalign{x_1+x_2+x_3+x_4&=0\cr2x_2+x_3+x_4&=0\cr4x_2+2x_3+2x_4&=0\cr} Do you know how to find the solutions of such a homogeneous system of linear equations? Do you know how to find a basis for that set of solutions? If not, better learn it fast, as you'll be using that technique over and over and over in linear algebra.