Kernel of Linear Transformation

Question

I have this question about finding the kernel of this linear transformation:

$L : \mathbb{R}^3 \to \mathbb{R}^3$ defined by: $$L : (x, y, z) \mapsto (x + y + z, 2x, 2x − y − z).$$

I have no idea how to start this question, I tried searching for videos and tutorials but nothing there. I have got exam tomorrow, any help would be appreciated.

Thanks in advance.

Answer 1

Solve for the system $L(x,y,z)=(0,0,0)$, using Gauß's pivot method.

Answer 2

The kernel of a linear transformation $T : \mathbb{R}^n \to \mathbb{R}^m$ is the set of vectors $\bar{x}$ in $\mathbb{R}^n$ such that $T(\bar{x}) = \bar{0}$.

In your case, you want to find real numbers $x,y,z$ so that $$x + y + z = 2x = 2x - y - z = 0.$$ Now $2x = 0$ implies $x = 0$. So then you are really just solving for $y,z$ so that $y + z = 0$ and $-y - z = 0$. Try to finish from here.