The null space of A is the plane x+y=0 Find the standard matrix $A$ for the operator $T$, given that the null space of $A$ is the plane $x+y=0$ and $T(1,0,0)=(1,1,0)$
The biggest issue I am having with this question is that I can't figure out how to turn the equation for the plane into something I can use to find the null space. Any help with this part would be appreciated. Thanks.
 A: First of all, the null space is of dimension $=2$, hence the rank of $T$ is $1$.
Now, if $x+y=0$, then $T(x,y,z)=0$. So what can you know about the matrix? Try to write out the matrix, first by letters $a, b, c$, and then substitute some vectors $(x,y,z)$ in, to find out the matrix.  

 Since $T(1,-1,z)=0, \forall z$,  we deduce that the third column of the matrix is $0$-column. Moreover, we deduce that $a_{12}=1, a_{22}=1$ in order the condition to be satisfied. Finally, the third row must be $O$-row, why?  

Inform me of any mistakes. Thanks in advance.
A: Hints:
$$x+y=0\iff y=-x\implies T\begin{pmatrix}x\\\!\!\!-x\\z\end{pmatrix}=A\begin{pmatrix}x\\\!\!\!-x\\z\end{pmatrix}=\begin{pmatrix}0\\0\\0\end{pmatrix}$$
Now, take $\,\{(1,0,0)\}\,$ and complete this to a basis of with two elements from the kernel of $\,A\,$, for example:
$$\begin{pmatrix}1\\\!\!\!-1\\0\end{pmatrix}\;,\;\;\begin{pmatrix}0\\0\\1\end{pmatrix}$$
and now express any vector as combination of this basis and apply $\,T\;(\text{or}\;A)\,$ keeping in mind this is a linear map.
Finally, express the standard basis as linear combination of the above basis and using the map find the coefficients that'll conform $\,A\,$
