Construction of a Linear Transformation question

Construction of a Linear Transformation question

Let there be a linear transformation going from $\mathbb R^4$ to $\mathbb
R^3$, such that $(1,0,1,0)$ , $(2,1,3,0)$ span ker T , and $(0,1,2)$ ,
$(3,1,2)$ span Im T. If such a transformation possible , construct it.
Attempt at a Solution: after Gaussian elimination on the span of the
kernel, I got $(1,0,1,0)$ and $(0,1,1,0)$. given the fact these two span
the kernel, after performing the transformation on them we get the zero
vector. These two then need to be expanded to $\mathbb R^4$, and here is
my question...I was told to choose the vectors $(0,0,1,0)$ and $(0,0,0,1)$
. Is it mandatory for these two vectors to be used in expansion or any
vectors independent on each other and the kernel spanning vectors would
do?
After this, I assign the vectors spanning the Im as results of the
transformation for the last two vectors discussed....from there I perform
the whole transformation on a vector of $\mathbb R^4$ , using the bases,
and reach the transformation matrix required.