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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Spring 2020 • Question 6

Question

Math 6310 • Algebra Spring 2020 • Question 6

Let \(n\) be a positive integer, \(V\) a \(\mathbb {C}\)-vector space of rank \(n\), and \(T:V\longrightarrow V\) a \(\mathbb {C}\)-linear map with the property that for each \(c\in \mathbb {C}\) the eigen space \(\{v\in V\mid T(v)=cv\}\) has rank at most 1. Prove that there exists a \(w\in V\) such that the set \(\{w,T(w),\ldots ,T^{n-1}(w)\}\) is a basis for \(V\).

Raw LaTeX

Let \(n\) be a positive integer, \(V\) a \(\mathbb{C}\)-vector space of rank \(n\), and
\(T:V\longrightarrow V\) a \(\mathbb{C}\)-linear map with the property that for each
\(c\in\mathbb{C}\) the eigen space \(\{v\in V\mid T(v)=cv\}\) has rank at most 1.

Prove that there exists a \(w\in V\) such that the set
\(\{w,T(w),\ldots,T^{n-1}(w)\}\) is a basis for \(V\).

Topic Tags

Matrix Similarity, Jordan Forms, and Finite-Order Diagonalization