# Thread: justifying that a definition is "well-defined"

1. ## 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

2. Originally Posted by squarerootof2
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.