# Thread: [SOLVED] Linear Algebra, basis of eigenvectors

1. ## [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 $\displaystyle T$.

2. 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