# [SOLVED] Linear Algebra, basis of eigenvectors

• Apr 21st 2010, 11:43 AM
JJMC89
[SOLVED] Linear Algebra, basis of eigenvectors
Let \$\displaystyle V\$ be a finite dimensional vector space over a field \$\displaystyle \mathbb{F}\$. Let \$\displaystyle T\$ be a linear
transformation on \$\displaystyle V\$ satisfying \$\displaystyle T^{2} = T\$. Show that \$\displaystyle V\$ admits a basis of eigenvectors
for \$\displaystyle T\$.
• Apr 21st 2010, 01:01 PM
tonio
Quote:

Originally Posted by JJMC89
Let \$\displaystyle V\$ be a finite dimensional vector space over a field \$\displaystyle \mathbb{F}\$. Let \$\displaystyle T\$ be a linear
transformation on \$\displaystyle V\$ satisfying \$\displaystyle T^{2} = T\$. Show that \$\displaystyle V\$ admits a basis of eigenvectors
for \$\displaystyle T\$.

\$\displaystyle T^2=T\iff T(T-I)=0 \Longrightarrow \$ the minimal polynomial of \$\displaystyle T\$ divides \$\displaystyle x(x-1)\Longrightarrow\$ the min. pol. of \$\displaystyle T\$ is the product of different linear factors \$\displaystyle \iff T\$ is

diagonalizable \$\displaystyle \iff \$ there's a basis of \$\displaystyle V\$ of eigenvectors of \$\displaystyle T\$.

Tonio