Suppose the 3 by 3 matrix A is invertible Write down bases f

Suppose the 3 by 3 matrix A is invertible. Write down bases for the four subspaces of A, and also for the 3 by 6 matrix B = [A A] (i.e., two copies of A placed side by side).

Solution

We have given that 3 by 3 matrix is invertible. Now, The row and column space of A are both R3, so a basis for those is (1, 0, 0), (0, 1, 0), and (0, 0, 1). The nullspace and the left nullspace of A are both zero, so the bases for those are empty. The column space of B is the same as the column space of A, so the same basis
works there, the row space has basis (1, 0, 0, 1, 0, 0), (0, 1, 0, 0, 1, 0), and (0, 0, 1, 0, 0, 1). The left
nullspace of B is zero, so its basis is empty, and the nullspace of B has basis (1, 0, 0, 1, 0, 0),
(0,1, 0, 0, 1, 0) and (0, 0,1, 0, 0, 1).