[Math] Invariant Subspace containing linear combination of eigenvectors

Question

Let
$$T:V\to V$$
be a linear transformation. Suppose that $v_1, v_2, \cdots, v_k \in V$ are eigenvectors of $T$ that correspond to distinct eigenvalues. Assume that $W$ is a $T$-invariant subspace of $V$ that contains the vector $v_1 + v_2 + \cdots + v_k$. Show that $W$ contains each of $v_1, v_2, \cdots, v_k$.

Answer 1

Let $\lambda_i$ be the eigenvalue associated to $v_i$.

Proof by induction. For $k=1$ this is trivial. Now assume that $v_1+...+v_k\in W$. But then also $\lambda_1v_1+...+\lambda_kv_k\in W$ by $T$-invariance and $\lambda_1v_1+...+\lambda_1v_k\in W$. Hence $(\lambda_2-\lambda_1)v_2+...+(\lambda_k-\lambda_1)v_k\in W$. Now apply the induction hypothesis on $(\lambda_2-\lambda_1)v_2,...,(\lambda_k-\lambda_1)v_k$ and receive $v_2,...,v_k\in W$. But then of course $v_1\in W$ aswell.