Eigenvalues and subspaces

Question

Suppose that $v_1 \neq v_2 \neq ... \neq v_n$ are eigenvectors of a matrix $A$, $n>3$. We know that eigenvectors form a subspace of $R^n$.

But is it true to say that, if we take a subset of these, for example $\{v_1,v_2,v_3\}$, span a subspace of $R^n$ of dimension $3$?

Answer 1

The (non-zero) eigenvectors corresponding to different eigenvalues are linearly independent.

Thus, the span of $k$ (non-zero) eigenvectors corresponding to $k$ different eigenvalues will be of dimension $k$.

Answer 2

Hint:

The eigenvectors of different eigenvalues are linearly indpendent. For aproof see:How to prove that eigenvectors from different eigenvalues are linearly independent