I recently teach myself linear algebra with Friedberg's textbook.
And I have a question about adjoint operator, which is on p.367.
Definition Let T : V → W be a linear transformation where V and W are finite-dimensional inner product spaces with inner products <‧,‧> and <‧,‧>' respectively. A funtion T* : W → V is called an adjoint of T if <T(x),y>' = <x,T*(x)> for all x in V and y in W.
Then ,my question is how to prove that there is a unique adjoint T* of T ?
Can anyone give me some tips ? thanks^^
That was much simpler than what I was thinking: You know there are 4 main properties of the adjoint; Suppose S, T are linear operators on V and let then
If you examine all of these properties, you will notice that the uniqueness of the adjoint implies that particiular property. For example , . You see the uniqueness of the adjoint implies (S+T)*=S*+T*.
Also, , again the uniqueness of the adjoint implies . The same with other two properties, they all show the uniqueness of the adjoint.