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 be a linear map between two finite-dimensional vector spaces, represented by a matrix of rank , then: is the dimension of the column space of , which represents the image of ; is the dimension of the null space of , which represents the kernel of ; is the dimension of the cokernel of . The transpose of is the matrix of the dual of . It follows that one has also: is the dimension of the row space of , which represents the image of ; is the dimension of the left null space of , which represents the kernel of ; is the dimension of the cokernel of . The two first assertions are also called the rank–nullity theorem. References Strang, Gilbert. Linear Algebra and Its Applications. 3rd ed. Orlando: Saunders, 1988. External links , MIT Linear Algebra Lecture on the Four Fundamental Subspaces, from MIT OpenCourseWare Theorems in linear algebra Isomorphism theorems