# Fundamental theorem of linear algebra

Short description: Name for certain results on linear maps between two finite-dimensional vector spaces

In mathematics, the fundamental theorem of linear algebra is a collection of statements regarding vector spaces and linear algebra, popularized by Gilbert Strang. The naming of these results is not universally accepted.

More precisely, let f be a linear map between two finite-dimensional vector spaces, represented by a m×n matrix M of rank r, then:

• r is the dimension of the column space of M, which represents the image of f;
• nr is the dimension of the null space of M, which represents the kernel of f;
• mr is the dimension of the cokernel of f.

The transpose MT of M is the matrix of the dual f* of f. It follows that one has also:

• r is the dimension of the row space of M, which represents the image of f*;
• mr is the dimension of the left null space of M, which represents the kernel of f*;
• nr is the dimension of the cokernel of f*.

The two first assertions are also called the rank–nullity theorem.