Adjoint Operator: Proving Unique Adjoint Transformation

[typhoonss821]

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^^

 

[radou]

Assume that there is another adjoint transformation, let's say T**.

 

[Mandark]

To show that there exists such a function, let $$v_1, \ldots , v_n$$ be an orthonormal basis for V, so that $$x = \sum_i \langle x,v_i\rangle v_i$$ for any x in V then we have for all x in V and y in W:

$$\langle T(x), y\rangle ' = \langle T (\sum_i \langle x,v_i\rangle v_i ), y\rangle '$$
$$= \sum_i \langle x, v_i\rangle \langle T(v_i), y\rangle '$$
$$= \langle x, \sum_i \overline{ \langle T(v_i),y\rangle '} v_i\rangle$$
which is in the form that we'd like.

Which shows that $$T^*(y) = \sum_i \overline{\langle T(v_i),y\rangle '} v_i$$ for all y in W works.

 

[typhoonss821]

really appreciate^^

 

[blue_raver22]

I can provide some tips to help you prove the uniqueness of the adjoint operator.

1. Understand the definition: Before attempting to prove the uniqueness of the adjoint operator, it is important to fully understand the definition. Make sure you are clear on what the adjoint operator is and what properties it must satisfy.

2. Use the properties of inner products: The definition of the adjoint operator involves inner products in both the domain and codomain. Use the properties of inner products, such as linearity and symmetry, to manipulate the equation and see if you can arrive at a conclusion about the uniqueness of the adjoint operator.

3. Use the properties of linear transformations: The linear transformation T and its adjoint T* both have specific properties that must hold. Use these properties, such as linearity and the fact that T* is the inverse of T, to see if you can prove that there can only be one adjoint operator for a given linear transformation.

4. Use a proof by contradiction: If you are having trouble proving the uniqueness of the adjoint operator directly, you can try a proof by contradiction. Assume that there are two different adjoint operators for a given linear transformation, and then use the properties of inner products and linear transformations to show that this assumption leads to a contradiction.

5. Consult other resources: If you are still struggling to prove the uniqueness of the adjoint operator, don't be afraid to consult other resources such as textbooks, online lectures, or asking for help from a professor or fellow student. Sometimes a different perspective can help you see the proof in a clearer way.

Remember, proving the uniqueness of the adjoint operator requires a solid understanding of linear algebra and the properties of inner products and linear transformations. Take your time, work through the equations, and don't be afraid to ask for help if needed. Good luck!