# Math Help - [SOLVED] Linear Algebra, basis of eigenvectors

1. ## [SOLVED] Linear Algebra, basis of eigenvectors

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

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

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

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

Tonio