Given linearly dependent set $U=\{x, Bx, B^2x,..,B^nx\}$ where $B$ is $n \times n$ matrix prove that matrix $k_0I+k_1B+..+k_nB^n$ is not invertible.
How to prove that $c_0I+c_1A+..+C_nA^n$ is singular if $(v,Av,A^2v,...,A^nv)$ is linearly dependent, $v \in \mathbb{F^n}$ and $A_{n \times n}$?