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Adjoint operator 
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#1
Feb410, 03:42 AM

P: 14

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 finitedimensional 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^^ 


#2
Feb410, 04:02 AM

HW Helper
P: 3,224

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



#3
Feb410, 05:07 AM

P: 41

To show that there exists such a function, let [tex]v_1, \ldots , v_n[/tex] be an orthonormal basis for V, so that [tex]x = \sum_i \langle x,v_i\rangle v_i[/tex] for any x in V then we have for all x in V and y in W:
[tex]\langle T(x), y\rangle ' = \langle T (\sum_i \langle x,v_i\rangle v_i ), y\rangle '[/tex] [tex] = \sum_i \langle x, v_i\rangle \langle T(v_i), y\rangle '[/tex] [tex] = \langle x, \sum_i \overline{ \langle T(v_i),y\rangle '} v_i\rangle [/tex] which is in the form that we'd like. Which shows that [tex]T^*(y) = \sum_i \overline{\langle T(v_i),y\rangle '} v_i[/tex] for all y in W works. 


#4
Feb410, 06:04 AM

P: 14

Adjoint operator
really appreciate^^



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