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In its most basic form, the rank nullity theorem states that for the [[Linear_transformation|linear transformation]] T represented by the m by n matrix A, then <math>\text{rank}(A)+\text{nullity}(A)=m</math>. 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_(linear_algebra)|kernel]] of A and the rank is the dimension of the [[Image_(linear_algebra)|image]] of A. Therefore the rank nullity theorem can be re-written as <math>\text{dim}(\text{im}(A))+\text{dim}(\text{ker}(A))=m</math> where <math>\text{im}(A)</math> is the [[Image_(linear_algebra)|image of the matrix]] A, and <math>\text{ker}(A)</math> is the [[Kernel_(linear_algebra)|kernel of the matrix]] A. | In its most basic form, the rank nullity theorem states that for the [[Linear_transformation|linear transformation]] T represented by the m by n matrix A, then <math>\text{rank}(A)+\text{nullity}(A)=m</math>. 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_(linear_algebra)|kernel]] of A and the rank is the dimension of the [[Image_(linear_algebra)|image]] of A. Therefore the rank nullity theorem can be re-written as <math>\text{dim}(\text{im}(A))+\text{dim}(\text{ker}(A))=m</math> where <math>\text{im}(A)</math> is the [[Image_(linear_algebra)|image of the matrix]] A, and <math>\text{ker}(A)</math> is the [[Kernel_(linear_algebra)|kernel of the matrix]] A. | ||
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Revision as of 04:53, 18 August 2010
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.
