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How can I define a linear operator $T:V_3\to V_3$ such that $\ker(T)=\operatorname{span}\{{(1,1,1)}\}$ and $\operatorname{Im}(T)=\operatorname{span}\{{(1,2,0), (2,3,-1)}\}$

I have no idea of what to do.

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Find yourself a $3\times3$ matrix in which two of the columns are $(1,2,0)$ and $(2,3,-1)$, and then fill in the third column so that when you multiply the matrix by $(1,1,1)$ you get the zero vector. That matrix will be the one representing $T$.

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