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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.

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  • $\begingroup$ errr.... since I'm all new to this I don't even know how to give it a start lol $\endgroup$ – Muhammad Qureshi Nov 26 '15 at 21:57
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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.

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  • $\begingroup$ I got y+z=0 and -y=z; Any clue whats the next step from here? :/ $\endgroup$ – Muhammad Qureshi Nov 26 '15 at 22:21
  • $\begingroup$ That essentially is the answer. Just write it in vector form, i.e. $$\ker L = \{(0,t,-t) : t \in \mathbb{R}\}.$$ $\endgroup$ – Ethan Alwaise Nov 27 '15 at 2:18
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Solve for the system $L(x,y,z)=(0,0,0)$, using Gauß's pivot method.

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