Find a Jordan basis $\beta$ for $T$ and calculate $[T]_{\beta, \beta}$And calculate det$(T)$ and trace($T)$

Question

Suppose $V$ is a vector of dimension $4$ over $\mathbb{R}$ and $T:V\to V$ is a linear transformation and then there exist nonzero vectors $v_1,v_2,v_3,v_4$ such that $$T(v_1)=7v_1+v_2, \;\;T(v_2)=7v_2, \;\;(T-4I)(v_3)=0, \;\;T(v_4)=4v_4$$ Find a Jordan basis $\beta$ for $T$ and calculate $[T]_{\beta, \beta}$And calculate det$(T)$ and trace($T)$

I really have no idea to where to start ..can you some one help me

Answer 1

Hint

The third equation can be written as: $$T (v_3)=4 v_3$$ Then using this what happens with $\beta = (v_1,v_2,v_3,v_4) $ ?