nLab fundamental theorem of linear algebra

Fundamental theorem of linear algebra

see also rank-nullity theorem

Fundamental theorem of linear algebra

Statement in operator form

For f:VWf \colon V\to W a linear operator between vector spaces, where VV is finite-dimensional of dimension nn, then the dimensions of the kernel and the image of ff are related by:

dimKerf+dimImf=n. dim\,Ker\,f + dim\,Im\,f \;=\; n \,.

Statement in the matrix form

If AA is a finite rectangular matrix then the row rank (the dimension of the span of its rows (“row space”)) plus the dimension of the solution space of equation Ax=0A x = 0 (“nullspace”) yields the number of columns. From the other side, column rank (the dimension of the space of columns) plus the dimension of the solution space of equation yA=0y A = 0 (or, equivalently, A Ty T=0A^T y^T = 0) yields the number of rows.

Notice that now we have two statements as a rectangular matrix can be thought as a linear operator in two ways: acting by left multiplication on column vectors or acting by right multiplication on row vectors.

Terminology

Sometimes additional statements are included by some authors.

Most often the additional part quoted is that the row rank is the same as the column rank. This is a somewhat nontrivial statement in our formulation (dimension of the space of rows/columns); in some expositions of the theory the definitions of ranks are such that this statement is automatic. Namely, the rank is the minimal dimension rr such that the m×nm\times n-matrix AA can be written in the form VΛ TV \Lambda^T where Λ\Lambda is n×rn\times r, that is Λ T\Lambda^T is r×nr\times n, and VV is an m×rm\times r matrix. Then the transposed matrix can be written as ΛV T\Lambda V^T. This is nothing than the image factorization of a linear map and its transpose which both go through the same intermediate space.

Literature

The following 2 refereces have somewhat nonstandard exposition of linear algebra to be amenable to constructive/algorithmic treatments useful in numerical linear algebra:

  • Carl de Boer?, Linear algebra, pdf
  • Carl de Boer?, Numerical functional analysis, University of Wisconsin lecture notes, chapter 1 (linear algebra), pdf
category: algebra

Last revised on May 21, 2024 at 13:17:32. See the history of this page for a list of all contributions to it.