I have the following polynomial
$$p(t) = a_0 + a_1 t + a_2 t^2 + t^3$$
and the following information
Let $p$ be indicated as above. Suppose $\lambda$ is a real root in the polynomials p, in other words, λ is a real eigenvalue for C. Let Eλ be the corresponding space for C; then $ Eλ = {x∈Rn | Cx = λx}$
I have been given three questions
(i) Define the Companion matrix for the polynomial above.
The first question is answered which corresponds to the matrix bellow
$ C_p=\begin{bmatrix} 0 & 1 & 0\quad\\ 0 & 0 & 1\quad\\ -a_0 & -a_1 & -a_2\quad\\ \end{bmatrix}$
(ii) Show that $\quad$ $ v_λ =\begin{bmatrix} 1\\\ λ^2\\ λ^3\\ \end{bmatrix}$ is an eigenvector for C which belongs to λ
That question is answered to0. Since λ is a zero of p, then $ a_0 + a_1λ+ a_2λ^2 +λ^3$ and $ -a_0 -a_1λ -a_2λ^2 = λ^3 $ gives us
$ C_p =\begin{bmatrix} 1\\\ λ^2\\ λ^3\\ \end{bmatrix}$ = $ =\begin{bmatrix} λ\\\ λ^2\\ -a_0 -a_1λ - a_2λ^2\\ \end{bmatrix}$ = $ =\begin{bmatrix} λ\\\ λ^2\\ λ^3\\ \end{bmatrix}$
That is $C_p(1,λ,λ^2) = λ(1,λ,λ^2)$ which shows that $(1,λ,λ)$ is an eigenvector of C_p corresponding to the eigenvalue λ $
(iii) Explain that $E_λ =Span (v_λ)$ and therefore $ dim E_λ = 1$
How do I solve the last question using what I have done so far?