# ahw4 - Math 5126 Monday, February 11 Fourth Homework...

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Math 5126 Monday, February 11 Fourth Homework Solutions 1. Section 12.2, Exercise 18 on page 489. Let V be a ﬁnite dimensional vector space over Q and suppose T is a nonsingular linear transformation of V such that T - 1 = T 2 + T . Prove that the dimension of V is divisible by 3. If the dimension of V is precisely 3, prove that all such transformations T are similar. Multiplying the given equation by T , we obtain T 3 + T 2 - I = 0 (where I is the identity matrix) and hence the minimal polynomial of T divides x 3 + x 2 - 1. Since x 3 + x 2 - 1 is irreducible over Q (if it wasn’t, it would have a degree one factor, which must be of the form x - a with a Z , hence ± 1 is a root of x 3 + x 2 - 1, which is not the case), it must be the minimal polynomial over Q . Therefore the only possibility for the invariant factors is x 3 + x 2 - 1 , x 3 + x 2 - 1 ,... . This tells us that the dimension of V must be a multiple of 3. If in addition the dimension of V is exactly 3, then the only possibility for the invariant factors is x 3 + x 2 -

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## This note was uploaded on 01/29/2009 for the course MATH 5125 taught by Professor Palinnell during the Fall '07 term at Virginia Tech.

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ahw4 - Math 5126 Monday, February 11 Fourth Homework...

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