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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Spring 2004 • Question 3

Question

Math 6310 • Algebra Spring 2004 • Question 3

Let \(V\) be a finite dimensional vector space over the field of rational numbers and let \(T\) be a linear transformation of \(V\) into itself such that \(T^5\) is the identity mapping. If there is no nonzero vector in \(V\) which \(T\) maps to itself, prove that the dimension of \(V\) is a multiple of 4.

Raw LaTeX

Let \(V\) be a finite dimensional vector space over the field of rational numbers and let \(T\) be
a linear transformation of \(V\) into itself such that \(T^5\) is the identity mapping. If there is
no nonzero vector in \(V\) which \(T\) maps to itself, prove that the dimension of \(V\) is a multiple
of 4.

Topic Tags

Finite Fields Matrix Similarity, Jordan Forms, and Finite-Order Diagonalization