Show Linear Independence of Set with T:V->V Operator

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SUMMARY

The discussion focuses on demonstrating the linear independence of the set {v, T(v), ..., T^(m-1)(v)} where T: V -> V is a linear operator on a vector space V over a field F. It is established that if v is a non-zero vector and T^m(v) equals zero while all previous applications of T yield non-zero results, then the set is linearly independent. The approach involves applying the operator T to the linear combination of the vectors and analyzing the implications of the resulting equations.

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  • Understanding of linear operators in vector spaces
  • Familiarity with the concept of linear independence
  • Knowledge of the properties of vector spaces over fields
  • Experience with applying operators repeatedly in linear algebra
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Students and professionals in mathematics, particularly those studying linear algebra, vector spaces, and linear transformations. This discussion is beneficial for anyone looking to deepen their understanding of linear independence and operator theory.

mivanova
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Can you help me with this, or at least give me an idea how to proceed:
Let T:V->V be a linear operator on the vector space over the field F. Let v is in V and let m be a positive integer for which v is not equal to 0, T(v) is not equal to 0, ...,T^(m-1)(v) is not equal to 0, but T^m(v) is equal to 0. Show that {v, T(v), ... , T^(m-1)(v)} is linearly independent set.
Thank you!
 
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Given a_0 v + a_1 T(v) + \cdots + a_{m-1} T^{m-1}(v) = 0, apply T to both sides repeatedly and see what you get.
 
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