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February 6th, 2012, 11:42 AM   #1
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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.
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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.
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