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;
• n – r is the dimension of the null space of M, which represents the kernel of f;
• m – r is the dimension of the cokernel of f.

The transpose M^T 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^*;
• m – r is the dimension of the left null space of M, which represents the kernel of f^*;
• n – r is the dimension of the cokernel of f^*.

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

References

• Strang, Gilbert. Linear Algebra and Its Applications. 3rd ed. Orlando: Saunders, 1988.
• Strang, Gilbert (1993), "The fundamental theorem of linear algebra", American Mathematical Monthly 100 (9): 848–855, doi:10.2307/2324660, http://www.dm.unibo.it/~regonati/ad0708/strang-FTLA.pdf 
• Banerjee, Sudipto; Roy, Anindya (2014), Linear Algebra and Matrix Analysis for Statistics, Texts in Statistical Science (1st ed.), Chapman and Hall/CRC, ISBN 978-1420095388 

External links

• Gilbert Strang, MIT Linear Algebra Lecture on the Four Fundamental Subspaces, from MIT OpenCourseWare