Prove result about basis of a linear map with specific properties I am working on the following problem.
Let $V$ be an $n$-dimensional vector space over $K$ and $T: V\to V$ a linear map.
For $k = 1, \ldots, n$ let $x_k \in V \smallsetminus \{0\}$ and $\lambda_k \in K$ be given such that $T(x_k) = \lambda_k x_k \space \forall k.$
Assume that $\lambda_k \not= \lambda_l$ whenever $k \not= l.$
Prove that $(x_1, \ldots, x_n)$ is a basis of $V$.
Hint: assume $\sum_{k = 1}^n \alpha_k x_k = 0$, where $l:= \max\{k \mid \alpha_k \not= 0\}$ is as small as possible.
Then apply $\lambda_l\text{Id}_V - T$ to both sides.
So far, I have managed to prove that, for $n >= 1, x_1, \ldots, x_n$ must be linearly independent (by applying the hint).
However, I am not yet sure how to show that $x_1, \ldots, x_n$ span $V$.
I would appreciate help with this.

If $V$ is a vector space of dimension $n$ over $K$, then any $n$ linearly independent vectors in $V$ form a basis of $V$.
Hence $x_1, \ldots, x_n$ span $V$.
