Let $$T: V\rightarrow V$$ $\quad$ $$V=\mathbb{R^3}$$ given the following orthonormal basis with respect to an inner product space (not the standard) $$B=\{(1,1,1),(1,1,0),(1,0,0)\}$$ I am sked to find a normal operator with the following charachteristic polynomial $$f_{T}(x)=x^{2}(x-3)$$ and also $$\operatorname{ker} T=\operatorname{Span}\{(3,2,1),(2,1,1)\}$$
I am asking to find an normal operator $T$ that will satisfy those conditions, and after that I am asked how many normal operators exists that satisfy those conditions.
so the first part I succeded and I found that the represent matrix $A$ with respect to the orthonormal basis B is: $$\begin{pmatrix}\frac{3}{2}&0&-\frac{3}{2}\\ 0&0&0\\ -\frac{3}{2}&0&\frac{3}{2}\end{pmatrix}$$
Now I do not know how to answer the second part, how many of those exists? how should I think on such question?