Linear algebra- Diagonalization of a symmetric matrix

A linear transformation $$T:R^3→R^3$$ is defined as $$T:(x)=Cx$$ where $$C$$ is a symmetric matrix.

a) State the dimensions of the eigenspaces $$\mbox{N(C-αI)}$$ and $$N(c-βI)$$

It is also given that: $$C\begin{pmatrix} -3 \\ 0 \\ 1 \\ \end{pmatrix} = \begin{pmatrix} -3α \\ 0 \\ α \\ \end{pmatrix}, C\begin{pmatrix} -2 \\ 1 \\ 0 \\ \end{pmatrix} = \begin{pmatrix} 2α \\ α \\ 0 \\ \end{pmatrix}$$

It is given that $$C$$ has a repeated eigenvalue $$α$$ and another eigenvalue $$β$$, where $$α \neq β$$

b) Find a basis for $$\mbox{N(C-αI)}$$ and $$N(c-βI)$$

→I understand that if $$C$$ is symmetric then $$C$$ has to be diagonalisable so that if $$α$$ is repeated the algebraic multiplicity= geometric multiplicity= 2, so $$N(C-αI)$$ is $$2$$ dimensional and $$N(c-βI)$$ is $$1$$ dimensional.

However, for part b), I don't understand how I would find the bases for the eigenspaces with the information I've been given.

• What is $\beta$? – Berci Feb 10 at 12:53
• I had omitted an important sentence from the question, I added it in bold. Other than that I have no other information. – Jacopo Salvatore Feb 10 at 13:20