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Linear algebra(linear transformation)

  • Thread starter chuy52506
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  • #1
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Homework Statement


Let T be a Linear Transformation from U to V with dim(V)=m where m<infinity.



Homework Equations


Show that rank(T) less than or equal to m.


The Attempt at a Solution


Im really not that familiar with what rank or m is.
 

Answers and Replies

  • #2
dextercioby
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Well, familiarity depends on your level of involvement. So look for the definition of <rank> in your notes/textbook. You can't find it (though i doubt it), search the internet: planetmath.org, wikipedia.org, wolfram sites offer a wealth of information on linear algebra.

m is the dimension of the vector space V. It's maximum number of linear independent vectors in V.
 
  • #3
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ok so rank is the max. number of linearly independent rows or columns.
I was trying to manipulate rank(T)=dim(range(T)) im not sure if this is the correct starting point?
 
  • #4
dextercioby
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It is. What's range(T) equal to ?
 
  • #5
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No nevermind that belonged to another problem. Ok so i just have rank(T)+nullity(T)=m
 
  • #7
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It does not say what the range(T) is
 
  • #8
dextercioby
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Of course not, you have to know that from the beginning of the class on linear algebra. The range is a subset of V obtained by taking all possible images through T of all the elements of U (assumed to be equal to the domain of T). The range is also known as codomain of a linear map/operator.

If U is a linear space and T is a linear operator, one can easily show that range(T) is also a linear space, namely a vector subspace of V.
 
  • #9
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I think the rank-nullity theorem is exactly what you want to use.
 

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