Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

4 Issues per year


SCImago Journal Rank (SJR) 2016: 0.207
Source Normalized Impact per Paper (SNIP) 2016: 0.315

Open Access
Online
ISSN
1898-9934
See all formats and pricing
More options …
Volume 15, Issue 3 (Jan 2007)

Issues

The Rank+Nullity Theorem

Jesse Alama
Published Online: 2008-06-09 | DOI: https://doi.org/10.2478/v10037-007-0015-6

The Rank+Nullity Theorem

The rank+nullity theorem states that, if T is a linear transformation from a finite-dimensional vector space V to a finite-dimensional vector space W, then dim(V) = rank(T) + nullity(T), where rank(T) = dim(im(T)) and nullity(T) = dim(ker(T)). The proof treated here is standard; see, for example, [14]: take a basis A of ker(T) and extend it to a basis B of V, and then show that dim(im(T)) is equal to |B - A|, and that T is one-to-one on B - A.

  • [1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Google Scholar

  • [2] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Google Scholar

  • [3] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Google Scholar

  • [4] Grzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions. Formalized Mathematics, 5(4):485-492, 1996.Google Scholar

  • [5] Czesław Byliński. Binary operations applied to finite sequences. Formalized Mathematics, 1(4):643-649, 1990.Google Scholar

  • [6] Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.Google Scholar

  • [7] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Google Scholar

  • [8] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Google Scholar

  • [9] Czesław Byliński. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521-527, 1990.Google Scholar

  • [10] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Google Scholar

  • [11] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Google Scholar

  • [12] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Google Scholar

  • [13] Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.Google Scholar

  • [14] Serge Lang. Algebra. Springer, 3rd edition, 2005.Google Scholar

  • [15] Robert Milewski. Associated matrix of linear map. Formalized Mathematics, 5(3):339-345, 1996.Google Scholar

  • [16] Michał Muzalewski. Rings and modules - part II. Formalized Mathematics, 2(4):579-585, 1991.Google Scholar

  • [17] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.Google Scholar

  • [18] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329-334, 1990.Google Scholar

  • [19] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115-122, 1990.Google Scholar

  • [20] Andrzej Trybulec. Function domains and Fránkel operator. Formalized Mathematics, 1(3):495-500, 1990.Google Scholar

  • [21] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9-11, 1990.Google Scholar

  • [22] Andrzej Trybulec. Tuples, projections and Cartesian products. Formalized Mathematics, 1(1):97-105, 1990.Google Scholar

  • [23] Wojciech A. Trybulec. Basis of vector space. Formalized Mathematics, 1(5):883-885, 1990.Google Scholar

  • [24] Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821-827, 1990.Google Scholar

  • [25] Wojciech A. Trybulec. Linear combinations in real linear space. Formalized Mathematics, 1(3):581-588, 1990.Google Scholar

  • [26] Wojciech A. Trybulec. Linear combinations in vector space. Formalized Mathematics, 1(5):877-882, 1990.Google Scholar

  • [27] Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569-573, 1990.Google Scholar

  • [28] Wojciech A. Trybulec. Operations on subspaces in vector space. Formalized Mathematics, 1(5):871-876, 1990.Google Scholar

  • [29] Wojciech A. Trybulec. Pigeon hole principle. Formalized Mathematics, 1(3):575-579, 1990.Google Scholar

  • [30] Wojciech A. Trybulec. Subspaces and cosets of subspaces in vector space. Formalized Mathematics, 1(5):865-870, 1990.Google Scholar

  • [31] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.Google Scholar

  • [32] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Google Scholar

  • [33] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.Google Scholar

  • [34] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.Google Scholar

  • [35] Mariusz Żynel. The Steinitz theorem and the dimension of a vector space. Formalized Mathematics, 5(3):423-428, 1996.Google Scholar

About the article


Published Online: 2008-06-09

Published in Print: 2007-01-01


Citation Information: Formalized Mathematics, ISSN (Online) 1898-9934, ISSN (Print) 1426-2630, DOI: https://doi.org/10.2478/v10037-007-0015-6.

Export Citation

This content is open access.

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Karol Pąk
Formalized Mathematics, 2011, Volume 19, Number 2
[2]
Karol Pąk
Formalized Mathematics, 2011, Volume 19, Number 2
[3]
Karol Pąk
Formalized Mathematics, 2008, Volume 16, Number 3
[4]
Karol Pąk
Formalized Mathematics, 2008, Volume 16, Number 4
[5]
Karol Pąk
Formalized Mathematics, 2008, Volume 16, Number 4
[6]
Jesse Alama
Formalized Mathematics, 2008, Volume 16, Number 1

Comments (0)

Please log in or register to comment.
Log in