What is the Adjoint of a Linear Operator?

[rsa58]

Homework Statement

T is a linear operator on a finite dimensional vector space. then N(T*T)=N(T). the null space are equal.

Homework Equations

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?

 

[rsa58]

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

 

[HallsofIvy]

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