Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Spring 2006 • Question 9

Question

Math 6310 • Algebra Spring 2006 • Question 9

Let \(V\) be a finite dimensional vector space over a field \(F\) and \(T:V\longrightarrow V\) be a linear transformation. Show that there is an integer \(n>0\) such that \[ V\cong \operatorname {Ker}(T^n)\oplus \operatorname {Im}(T^n). \]

Raw LaTeX

Let \(V\) be a finite dimensional vector space over a field \(F\) and
\(T:V\longrightarrow V\) be a linear transformation. Show that there is an integer \(n>0\) such that
\[
V\cong\operatorname{Ker}(T^n)\oplus\operatorname{Im}(T^n).
\]

Topic Tags

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