Question about linear mappings and inner product spaces

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Homework Statement



Hi, I am having difficulty with the following proof:

Let V be an inner product space (real of dimension n) with two inner products in V, <,> and [,]. Prove that there exists a linear mapping on V such that [L(x),L(y)] = <x,y> for all x,y in V.

I am stuck as to where to go with the proof. I know that I need to construct a linear mapping with the above property, however I'm not sure where to go from there. Any insight into this would be appreciated.
Thanks


Homework Equations





The Attempt at a Solution


Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
jbunniii
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Hint: let ##\{e_1,\ldots,e_n\}## be an orthonormal (with respect to ##\langle , \rangle##) basis for ##V##. It suffices to find a linear map ##L## such that
$$[L(e_i), L(e_j)] = \langle e_i, e_j\rangle = \delta_{ij}$$
One way to proceed from here is to reformulate the problem in terms of matrices.
 
  • #3
Fredrik
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There's a simple solution that involves two orthonormal bases, one that's orthonormal with respect to <,>, and one that's orthonormal with respect to [,].
 

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