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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  
Senior Member Joined: Jul 2011 Posts: 227 Thanks: 0  Re: Linear Transformations in Linear algebra Quote:
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.  

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