Exact sequences split
June 26, 2008  Posted by John Armstrong  Algebra, Linear Algebra
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[…] since every short exact sequence splits we have an isomorphism . This is the content of the ranknullity […]
Pingback by The RankNullity Theorem « The Unapologetic Mathematician  June 27, 2008 
[…] Today I just want to point out a dual proposition to the one I refined last week. At that time we stated that any linearly independent set can be expanded to a basis. This followed […]
Pingback by Spanning sets « The Unapologetic Mathematician  June 30, 2008 
[…] the system is inhomogenous in general, and as such it might not have any solutions. Since every short exact sequence of vector spaces splits we can write . Then the vector will have some component in the image of , and some component in […]
Pingback by The Index of a Linear Map « The Unapologetic Mathematician  July 22, 2008 
[…] the map allows us to break up as (since short exact sequences split). On the other hand, considering the map allows us to break up as . Exactness tells us that , […]
Pingback by The Euler Characteristic of an Exact Sequence Vanishes « The Unapologetic Mathematician  July 23, 2008 
[…] say we’re given a list of linearly independent vectors in . They must be a basis, since any linearly independent set can be completed to a basis, and a basis of must have exactly elements, which we already have. Then we can use the as the […]
Pingback by The General Linear Groups « The Unapologetic Mathematician  October 20, 2008 
[…] we know that some of the most important exact sequences are short exact sequences. We also saw that every short exact sequence of vector spaces splits. So does the same hold for representations? It turns out that no, they don’t always, and […]
Pingback by Do Short Exact Sequences of Representations Split? « The Unapologetic Mathematician  December 17, 2008 
[…] take a basis of . Since this is a linearly independent set spanning a subspace of , it can be completed to a basis for all of . Now we can use this basis of to write out the matrix of and use our formula from […]
Pingback by The Determinant of a Noninvertible Transformation « The Unapologetic Mathematician  January 14, 2009 
[…] also know that any linearly independent set can be expanded to a basis. In fact, we can also extend any orthonormal collection of vectors to an orthonormal basis. Indeed, […]
Pingback by Orthonormal Bases « The Unapologetic Mathematician  April 29, 2009 
[…] An important fact about the category of vector spaces is that all exact sequences split. That is, if we have a short exact […]
Pingback by Orthogonal Complements « The Unapologetic Mathematician  May 4, 2009 
[…] when we needed to show that every vector space has a basis, or Tychonoff’s theorem, or that exact sequences of vector spaces split. So it’s sort of a mixed bag. In practice, most working mathematicians seem to be willing to […]
Pingback by NonLebesgue Measurable Sets « The Unapologetic Mathematician  April 24, 2010 
[…] submodule — and — then we can pick a basis of . And then we know that we can extend this to a basis for all of : […]
Pingback by Reducibility « The Unapologetic Mathematician  September 23, 2010 