show that $I, T, T^2, ..., T^k$ are linear dependent

Question

I am learning linear algebra and new to it. I can not solve this problem. I think it has a trick that I don't know.

for T(a linear map), $T:V\rightarrow V$ and every $v$ in $V$ the $v, T(v), T^2(v), ..., T^k(v)$ are linearly dependent where $k$ is a natural number $\leq \dim(V)$.

show that $I, T, T^2, ..., T^k$ are linearly dependent

thanks in advance.

Answer 1

Suppose $a_1T+...+a_kT^k=0$. Let $x\neq 0$, $a_1T(x)+...+a_kT^k(x)=0$, this implies that $a_1=..=a_k=0$ since $T(x),...T^k(x)$ are linearly independent. It results that $T,....T^k$ is linearly independent.

Remark, you have to suppose for every $v\neq 0$, $T(v),...,T^k(v)$ is linearly independent since it is not true for $v=0$.

Answer 2

Hint :

Suppose that $T,T^2,\dots, T^k$ are not linearly independent. Then, there exists $\lambda_1, \lambda_2, \dots, \lambda_k \in \mathbb R$ not all zero, such that :

$$\lambda_1T + \lambda_2T^2 + \cdots + \lambda_kT^k = 0$$