vector spaces – Find the linear map given image and kernel and a parameter

Let $W_{k}$ be a subspace and $k$ a real parameter.

$W_{k} = <(1,2,k),(1,k,1),(2,4,3)>$

I have to determine for what values of $k$ exist a linear map $F_{k}:mathbb{R^3} to mathbb{R^3}$ such that

$Im(F_{k})=W_k\ker(F_k)=<(1,2,2)>$

I found that for $k neq 2 $ and $k neq dfrac{3}{2}$ the vectors $(1,2,k),(1,k,1),(2,4,3)$ form a basis of $W_k$ and $W_k$ is basically $mathbb R^3$.

The associated matrix should have 3 rows and 3 columns.

Since $ dim(ker(F_k)) = 1$, $ dim(Im(F_k))$ must be $2$ since the domain is $mathbb R^3$.

The image must be equal to $W_k$, so I should take values of $k$ so that $dim(W_k)=2$.

For example I could take $k=2$ and the image would be generated by $(1,2,2)$ and $(1,2,1)$ that would be the first two columns of the matrix and to find the third column I need to assure that $F(1,2,2) = (0,0,0)$. Is that correct?