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.