Thanks for your answer. The third question is completely clear to me now. But for the first and second questions, are there no connections between eigenspace and column space?
Is it true that if an n by n matrix A has n-linearly independent eigenvectors, then it must also be invertible because these n-eigenvectors span n-space. But does this reasoning work the other way around: that is if A is invertible, does that imply n-linearly independent eigenvectors can be...