How to find the kernel of a linear transformation $T:\mathbb{R^3}\to\mathbb{R^3}$. 
How to find the kernel of the linear transformation $T:\mathbb{R^3}\to\mathbb{R^3}$ given by $T(x,y,z)=(x,2y,0)$?

I do not quite understand how to do this!
and

How to find the kernel of the linear transformation $T:\mathbb{R^3}\to\mathbb{R^2}$ given by $T(x,y,z)=(x,y,z)\begin{bmatrix}1&2\\0&-1\\1&1\end{bmatrix}$?


To find the kernel of the first linear transformation, you must determine for which $(x,y,z)\in\Bbb R^3$ we have $T(x,y,z)=(0,0,0).$ I will leave that to you.
For the second linear transformation, we must determine $(x,y,z)\in\Bbb R^3$ for which $T(x,y,z)=(0,0).$ In particular, note that $$T(x,y,z)=(x+z,2x-y+z),$$ so we need $x+z=0$ and $2x-y+z=0.$ The first equation holds precisely when $z=-x,$ so the second equation becomes $x-y=0$ by substitution, which means $y=x$. Hence, we are talking about the set of vectors of the form $(x,y,z)=(x,x,-x)=x(1,1,-1).$ Your kernel, then, is the subspace of $\Bbb R^3$ generated/spanned by $(1,1,-1)$.
