The null space, or kernel, of a *m* x *n* matrix **A** is the set (vector space) of all solutions **x** to the equation **Ax** = **0** (including **x** = **0**). The dimension of the null space is *r* - *n*, where *r* is the rank of **A**

If the null space of **A** is more than the set {0}, then the columns of **A** are dependent and **A** is not invertible. If the null space is {0} (and **A** is square), the columns are independent, and the matrix is surely invertible.