: Mathematics: rank-nullity theorem

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rank-nullity 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 $f$ ).
It follows that for linear transformations of vector spaces of equal finite dimension, either injectivity or surjectivity implies bijectivity.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Rank%E2%80%93nullity_theorem)

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