Will the basis of Nul(A) and Col(A) be linearly independent in a nxn matrix?

Question

Will the vector set containing both the basis of Nul(A) and the basis of Col(A) be linearly independent for a nxn matrix?

If not, in which cases are they?

Answer 1

In general the question makes no sense since the null space is a subspace of the domain and the column space is a subspace of the codomain.

Even when you consider them to be subspaces of the same space (essentially by using the standard coordinate system), the answer is "no". For the matrix $$ A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} $$ the vector $(1,0)$ spans both the kernel and the image.

(This is an example satisfying $$ \ker A = \ker A^2 $$ as in the comment from @SergeyGolovan .)