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Minimal polynomial and stabilizer

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Any explanations of the attached definitions is greatly appreciated. Thank you.

The definition of the minimal polynomial for the linear transformation....
the relationship between the cosets in G of the stabilizer....

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Please see the attachment.

(a) For linear transformation , we define

Then the minimal polynomial of is the monic polynomial which generates .
In another word, the minimal polynomial is the monic polynomial at the least
degree in .
For example, maps from to . ...

Solution Summary

This explains the definition of a minimal polynomial for a linear transformation and the relationship between cosets in the stabilizer.