Fundamental theorem of linear algebra

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In mathematics, the fundamental theorem of linear algebra makes several statements regarding vector spaces.[which?] These may be stated concretely in terms of the rank r of an m × n matrix A and its singular value decomposition:

First, each matrix ( has rows and columns) induces four fundamental subspaces. These fundamental subspaces are:

name of subspace definition containing space dimension basis
column space, range or image or (rank) The first columns of
nullspace or kernel or (nullity) The last columns of
row space or coimage or (rank) The first columns of
left nullspace or cokernel or (corank) The last columns of

Secondly:

  1. In , , that is, the nullspace is the orthogonal complement of the row space
  2. In , , that is, the left nullspace is the orthogonal complement of the column space.
The four subspaces associated to a matrix A.

The dimensions of the subspaces are related by the rank–nullity theorem, and follow from the above theorem.

Further, all these spaces are intrinsically defined—they do not require a choice of basis—in which case one rewrites this in terms of abstract vector spaces, operators, and the dual spaces as and : the kernel and image of are the cokernel and coimage of .

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