For the matrix A below, compute the dimension of the null space of A, dim (N (A))

Row reduce A,

So r = 3 for this matrix. Then

dim (N (A)) = n (A)  Definition NOM

= (n (A) + r (A)) − r (A)

= 5 − r (A)                                                                 Theorem RPNC

= 5 − 3                                                     Theorem CRN

= 2

We could also use Theorem BNS and create a basis for N(A) with n-r = 5-   3 = 2 vectors (because the solutions are described with 2 free variables) and arrive at the dimension as the size of this basis.