Fundamental theorem of linear algebra

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 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*;
  • 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" (PDF), American Mathematical Monthly, 100 (9): 848–855, CiteSeerX 10.1.1.384.2309, doi:10.2307/2324660, JSTOR 2324660
  • 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
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