Let L V rightarrow V be a linear mapping such that L2 O Sho

Let L: V rightarrow V be a linear mapping such that L^2 = O. Show that I - L is invertible. (I is the identity mapping on V.) Let L: V rightarrow V be a linear map such that L^2 + 2L + I = O. Show that L is invertible. Let L: V rightarrow V be a linear map such that L^3 = O. Show that I - L is invertible.

Solution

By definition of Inevitability:

A linear map T : V W is called invertible if there exists a linear map S : W V such that

T S = IW and

ST = IV ,

where IV : V V is the identity map on V and IW : W W is the identity map on W.

We say that S is an inverse of T.

a)

let L:v--> V be a linear mapping such that L^2=0, show that L is invertible (I is the identity mapping on V)

If L is injective, then we know that null T = {0}.

Hence by the dimension formula we have

dim range L = dim V dim null L = dim V.

Since range L V is a subspace of V , this implies that range L = V and hence L is surjective.

Since by assumption L is surjective, we have range L = V . Hence again by the dimension formula

dim null L= dim V dim range L = 0,

so that

null L = {0},

and hence T is injective. By Proposition linear map is invertible.