Rank–nullity theorem
In linear algebra, relation between 3 dimensions / From Wikipedia, the free encyclopedia
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"Rank theorem" redirects here. For the rank theorem of multivariable calculus, see constant rank theorem.
The rank–nullity theorem is a theorem in linear algebra, which asserts:
- the number of columns of a matrix M is the sum of the rank of M and the nullity of M; and
- the dimension of the domain of a linear transformation f is the sum of the rank of f (the dimension of the image of f) and the nullity of f (the dimension of the kernel of f).[1][2][3][4]
It follows that for linear transformations of vector spaces of equal finite dimension, either injectivity or surjectivity implies bijectivity.