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I'm having trouble understanding why for an n x n matrix A, in order to have eigenvalues, it must have linearly dependent columns (so that a nontrivial solution exists), but for the same A, in order to be diagonalizable, the columns must be linearly INdependent.

The basis for the eigenspace of the former would be the null space, but for the latter, the basis would be the column space since no free variables exist.