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Let V be a finite dimensional vector space over a field F. Let T be linear transformation on V
satisfying T^2 = T. Show that V admits basis of eigenvectors for T.
Solution
Given that T^2=T
T^2-T=0
T(T-1)=0
that means characteristic equatuon wl be x(x-1)=0
I.e two distinxt eigen values and which implies that T is daigonazble then eigen vectors
corresponding to distinct eigen values are mutualy linearly independent and hence form basis for
V ( since T has distinct (dimV ) eigen values .)

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Let V be a finite dimensional vector space over a field F. Let T be l.pdf

  • 1. Let V be a finite dimensional vector space over a field F. Let T be linear transformation on V satisfying T^2 = T. Show that V admits basis of eigenvectors for T. Solution Given that T^2=T T^2-T=0 T(T-1)=0 that means characteristic equatuon wl be x(x-1)=0 I.e two distinxt eigen values and which implies that T is daigonazble then eigen vectors corresponding to distinct eigen values are mutualy linearly independent and hence form basis for V ( since T has distinct (dimV ) eigen values .)