# Homework Help: Adjoint of a linear operator

1. Feb 19, 2010

### rsa58

1. The problem statement, all variables and given/known data
T is a linear operator on a finite dimensional vector space. then N(T*T)=N(T). the null space are equal.

2. Relevant equations

3. The attempt at a solution
this is my method, but its does not work if dim(R(T))=0. i'm only concerned with showing
N(T*T) $$\subseteq$$ N(T). let x beong to N(T*T) then <T*T(x),y>=0=<T(x),T(y)> for all y in the vector space. thus, if dim(R(T)) > 0 then there exists y such that T(y) is not equal to zero so T(x)=0.

any other methods out there?

2. Feb 19, 2010

### rsa58

okay i think i got it if dim(R(T))=0 then ofcourse x is in the null space of T.

3. Feb 20, 2010

### HallsofIvy

But that's only true for the trivial case, the 0 matrix.