Theorem

Let 𝑣₁, 𝑣₂, …, 𝑣_𝑘 be eigenvectors of a matrix 𝐴, and suppose that the corresponding eigenvalues 𝜆₁, 𝜆₂, …, 𝜆_𝑘 are distinct (all different from each other).

Then {𝑣₁, 𝑣₂, …, 𝑣_𝑘} is linearly independent

Proof

TODO see: https://textbooks.math.gatech.edu/ila/eigenvectors.html