(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

let T(V)=V be a linear map, where V is a finite-dimensional vector space. Then T^2 is defined to be the composite TT of T with itself, and similarly T^(i+1) = TT^i for all i >=1. Suppose Rank (T) = Rank (T^2)

2. Relevant equations

a) prove that Im(T) = Im(T^2)

b) for i>=1, let U_i(Im(T))=Im(T) be definied as the restriction of T^i to the subspace Im(T) of V. Show that U_i is nonsingular for all i

c) Deduce that Rank (T) = Rank (T^i) for all i >= 1

3. The attempt at a solution

a) since Rank (T) = Rank (T^2), then

dim( Im(T) ) = dim ( Im(T^2) )

since Im (T) = V and TT(V) = T(V) = V = Im(T^2)

so Im(T) = Im (T^2) because V=V

-does that make sense?-

b) U_i (Im(T)) = Im(T)

so U_i (V) = V

since V is mapped to itself, U_i has to be an identity matrix

and identity matrix has an inverse because its determinant is not zero,

so U_i is nonsingular

-does that make sense?-

c) since T is a linear map that V to itself, T^i (V) = V for all i>=1,

implies that image would be the same

hence dimension of image is the same

so the Rank (T) = Rank (T^i) for all i>=1

-again, does that make sense?-

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Vector space and mapping

**Physics Forums | Science Articles, Homework Help, Discussion**