Finding if a linear transformation exist

Question

I have to following question which I find hard to solve.

Is there a linear transformation from $\mathbb{R}^5$ to $\mathbb{R}^4$ such that:

$T:\mathbb{R}^5→\mathbb{R}^4$

Such that:

$\text{ker}T=\{(x,y,z,w)\in \mathbb{R}^5 \mid x=2y $ and $ z=2t=3w\}$

Define it on a basis of $\mathbb{R}^5$

Now I understand that I need to complete the basis to $\mathbb{R}^5$ with the standard basis, but I don't know how.

From there, after I found that basis, I can prove that a linear transformation exists (and only one, such that $T(Xi)=Yi$.

Your help is appreciated, thank you.

Answer 1

$T(x,y,z,t,w)=(x-2y, z-2t ,z-3w, 2t-3w )$