Rank Nullity Theorem
In its most basic form, the rank nullity theorem states that for the linear transformation T represented by the m by n matrix A, then $ \text{rank}(A)+\text{nullity}(A)=m $. Where rank is the number of rows in A with leading ones and nullity is the number of rows without leading ones. Nullity also happens to be the dimension of the kernel of A and the rank is the dimension of the image of A. Therefore the rank nullity theorem can be re-written as $ \text{dim}(\text{im}(A))+\text{dim}(\text{ker}(A))=m $ where $ \text{im}(A) $ is the image of the matrix A, and $ \text{ker}(A) $ is the kernel of the matrix A.
