If you have some vector $r_1,r_2,n_1,n_2,c_1,c_2,l_1,l_2 \in R^4$
a) Under what condition it will be $\{r_1,r_2\},\{n_1,n_2\},\{c_1,c_2\},\{l_1,l_2\}$ base of fundamental subspace $R(A^T),N(A),R(A),N(A^T)$ in that order, of some matrices $A\in M_4(\mathbb R)$?
b) Under condition that you get for a) give an example of such a matrix.
$\mathbb R^4=N(A)⊕R(A)^T$ and $\mathbb R^4=R(A)⊕N(A)^T$ so $N(A)\cap R(A)^T=\{0\}$ if we take some vector $x\in N(A)\cap R(A)^T$ since $x\in N(A)$ and $x\in R(A)^T$ then $x=\alpha_1 r_1+\alpha_2 n2, x=\beta1 n_1+\beta2 n_2$ $\alpha_1 r_1+\alpha_2 n2=\beta1 n_1+\beta2 n_2$ since $\alpha_1 n_1-\beta_1n_1-\beta_2 n_2+\alpha_2 n_2=0$ they must be linear independent and they must be orthogonal, that is the same for $\{c_1,c_2\},\{l_1,l_2\}$, I try to find matrix but I have no idea,do you have some idea?