词条 | Fundamental theorem of linear algebra | |||||||||||||||||||||||||
释义 |
}} In mathematics, the fundamental theorem of linear algebra makes several statements regarding vector spaces. Those statements may be given concretely in terms of the rank r of an {{nowrap|m × n}} matrix A and its singular value decomposition: First, each matrix ( has rows and columns) induces four fundamental subspaces. These fundamental subspaces are as follows:
Secondly:
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 . See also
References
| title = The fundamental theorem of linear algebra | url = http://www.eng.iastate.edu/~julied/classes/CE570/Notes/strangpaper.pdf | year = 1993 | author = Strang, Gilbert | journal = American Mathematical Monthly | volume = 100 | issue = 9 | pages = 848–855 | doi = 10.2307/2324660 | jstor = 2324660 | citeseerx = 10.1.1.384.2309 }}
External links
3 : Theorems in linear algebra|Isomorphism theorems|Fundamental theorems |
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