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Math Help - justifying that a definition is "well-defined"

  1. #1
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    justifying that a definition is "well-defined"

    part a of this question was to show that similar matrices have the same trace, which i was able to prove without much difficulty.

    part b says, how would one define the trace of a linear operator T on a finite dimensional vector space? and how would one show that this definition is well-defined?

    i dont get what the question means when it says show the definition is well-defined. only notion of well-defined i have is for a function where if a=b then f(a)=f(b) but i dont see how that would apply here. thanks for any inputs
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  2. #2
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    Quote Originally Posted by squarerootof2 View Post
    part a of this question was to show that similar matrices have the same trace, which i was able to prove without much difficulty.

    part b says, how would one define the trace of a linear operator T on a finite dimensional vector space? and how would one show that this definition is well-defined?

    i dont get what the question means when it says show the definition is well-defined. only notion of well-defined i have is for a function where if a=b then f(a)=f(b) but i dont see how that would apply here. thanks for any inputs
    Pick an ordered basis B and form the matrix [T]_B. Define \text{Tr}(T) = \text{Tr}([T]_B). Of course the question here is this well-defined? Meaning what happens if we pick a different basis B' do we still get the same result? The answer is yes because remember [T]_B and similar [T]_{B'} are similar to each other (in fact that similarity matrix is a transition matrix between B and B'). Therefore it does not depend on what basis you use.
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