Set of eigenvectors is linearly independent

[Fermat1]

I know eigenvectors corresponding to different eigenvalues are linearly independent but what about a set ${e_{1},...,e_{n}}$ of eigenvectors corresponding to different eigenvalues?

 

[HallsofIvy]

I don't understand your question because I don't see how the two parts of your question are different.

When you say

I know eigenvectors corresponding to different eigenvalues are linearly independent

Do you mean that two eigenvectors corresponding to two different eigenvalues

but what about a set ${e_{1},...,e_{n}}$ of eigenvectors corresponding to different eigenvalues?

but asking, "what if there are more than two?". One can show generally, "if, in a set of vectors, any two are independent (au+ bv= 0 only if a= b= 0 which is the same as saying that b is NOT a multiple of a and vice-versa) then all the vectors are independent."
One can prove that by first proving 'if $$u_1, u_2, ... u_n$$ are each independent of v, then so is $$a_1u_1+ a_2u_2+ ...+ a_nu_n$$ is independent of v for any numbers $$a_1$$, $$a_2$$, ..., $$a_n$$.