[SOLVED] Linear Algebra, basis of eigenvectors

**JJMC89** · April 21st 2010, 11:43 AM

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 $T$ .

**tonio** · April 21st 2010, 01:01 PM

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

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