Rank–nullity theorem

In mathematics, the rank–nullity theorem of linear algebra, in its simplest form, states that the rank and the nullity of a matrix add up to the number of columns of the matrix. Specifically, if A is an m-by-n matrix (with m rows and n columns) over some field, then

This applies to linear maps as well. Let V be a finite dimensional vector space, and W be a (not necessarily finite dimensional) vector space over some field and let T : V → W be a linear map. Then

or, equivalently,

where the rank of T is the dimension of the image of T and the nullity of T is the dimension of the kernel of T.

One can refine this statement (via the splitting lemma or the below proof) to be a statement about an isomorphism of spaces, not just dimensions.

More generally, one can consider the image, kernel, coimage, and cokernel, which are related by the fundamental theorem of linear algebra.

We give two proofs. The first proof uses notations for linear transformations, but can be easily adapted to matrices by writing T(x) = Ax, where A is m × n. The second proof looks at the homogeneous system Ax = 0 associated with an m × n matrix A of rank r and shows explicitly that there exist a set of n − r linearly independent solutions that span the null space of A. These proofs are also available in the book by Banerjee and Roy (2014).

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