Proving the Rank Equivalence of Adjoint Operators

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The discussion focuses on proving the rank equivalence of adjoint operators in finite-dimensional inner product spaces. Specifically, it establishes that for a linear transformation T: V → W, the rank of the adjoint operator T* is equal to the rank of T. The proof involves selecting orthonormal bases for both V and W, allowing the inner product matrices to be represented as identity matrices, and then expressing the adjoint relationship in matrix form to draw conclusions about their ranks.

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typhoonss821
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I have a question about the rank of adjoint operator...

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.

My question is how to prove that rank(T*) = rank(T)??

Can anyone give me some tips, thanks^^
 
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Take a basis B of V and a basis B' of W in which the matrix of <,> and <,>' are both the identity. That is to say, pick B <,>-orthonormal and B' <,>'-orthonormal. This is always possible by Gram-Schmidt.

Then write the equation <T(x),y>' = <x,T*(x)> in matrix form with respect of these basis and conclude.
 

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