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.

  • I think that it should be "n by m" matrix, meaning that rank(A) + nullity(A)= m (number of columns, not rows as it states here)!!

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Abstract algebra continues the conceptual developments of linear algebra, on an even grander scale.

Dr. Paul Garrett