For a linear map between finite-dimensional vector spaces, the dimension of the domain equals the rank of the map plus the dimension of its kernel (the nullity). It is a basic accounting identity underlying much of linear algebra.
Facts
StatementFor a linear map T from a finite-dimensional vector space V, rank(T) plus nullity(T) equals dim(V): the dimension of the image plus the dimension of the kernel equals the dimension of the domain. 1 Classification
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1. Rank-Nullity Theorem (Wikipedia)
Wikimedia Foundationlead section, matrix formulationQuote, lead section, matrix formulation
the number of columns of a matrix M is the sum of the rank of M and the nullity of M
Rank-nullity theorem (Wikipedia)
In Group: Linear Algebra and Matrix Theory, Article textQuote, In Group: Linear Algebra and Matrix Theory, Article text
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
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