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 February 6th, 2012, 11:42 AM #1 Member   Joined: Apr 2011 Posts: 36 Thanks: 0 Linear Transformations in Linear algebra What is the most tangible way to introduce linear transformations? Most books tend to start with a very abstract view which is off putting for some students. Is there a short and simple proof of the Nullity - Rank Theorem which claims that if T: U->V is a linear transformation then rank(T)+Nullity(T)=n where n is the n dimension vector space U. February 7th, 2012, 01:39 PM   #2
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Re: Linear Transformations in Linear algebra

Quote:
 Originally Posted by matqkks Is there a short and simple proof of the Nullity - Rank Theorem which claims that if T: U \to V is a linear map then rank(T)+Nullity(T)=n where n is the n dimension vector space U.
There're different ways to prove this. One way:
Let be a linear map. We have to prove:

You should know the quotient space and are isomorfic vector spaces and therefore they have the same dimension. So we have:

Applying the dimension formula: we obtain:

This dimension formula is the most general form. Assume you have a matrix. . We can associate a linear map defined as:

Try to figure out how this implies into rank(T)+nulity(T)=n. Tags algebra, linear, transformations Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post azelio Linear Algebra 0 October 19th, 2011 06:59 AM nuke Linear Algebra 3 April 14th, 2011 12:32 PM TsAmE Linear Algebra 0 October 9th, 2010 06:54 AM wontonsoup Linear Algebra 2 May 25th, 2009 04:35 PM ypatia Linear Algebra 1 March 2nd, 2009 06:28 PM

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