The Rank+Nullity Theorem

Jesse Alama

Formalized Mathematics (2007)

  • Volume: 15, Issue: 3, page 137-142
  • ISSN: 1426-2630

Abstract

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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.

How to cite

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Jesse Alama. "The Rank+Nullity Theorem." Formalized Mathematics 15.3 (2007): 137-142. <http://eudml.org/doc/267154>.

@article{JesseAlama2007,
abstract = {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.},
author = {Jesse Alama},
journal = {Formalized Mathematics},
language = {eng},
number = {3},
pages = {137-142},
title = {The Rank+Nullity Theorem},
url = {http://eudml.org/doc/267154},
volume = {15},
year = {2007},
}

TY - JOUR
AU - Jesse Alama
TI - The Rank+Nullity Theorem
JO - Formalized Mathematics
PY - 2007
VL - 15
IS - 3
SP - 137
EP - 142
AB - 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.
LA - eng
UR - http://eudml.org/doc/267154
ER -

References

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Citations in EuDML Documents

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  1. Jesse Alama, Euler's Polyhedron Formula
  2. Kazuhisa Nakasho, Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama, Rank of Submodule, Linear Transformations and Linearly Independent Subsets of Z-module
  3. Karol Pąk, Eigenvalues of a Linear Transformation
  4. Karol Pąk, Jordan Matrix Decomposition
  5. Karol Pąk, Linear Transformations of Euclidean Topological Spaces. Part II
  6. Karol Pąk, Linear Map of Matrices
  7. Karol Pąk, Linear Transformations of Euclidean Topological Spaces
  8. Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama, Torsion Part of ℤ-module

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