Let v &#x2208;<!-- ∈ --> T 2 </msub> ( V ) be a bilinear form over finite s

Dania Mueller

Dania Mueller

Answered question

2022-06-27

Let v T 2 ( V ) be a bilinear form over finite space V. Let T be a Linear transformation V V. We define: v T ( x , y ) = v ( T ( x ) , y )
Assuming v is nondegenerate, let us have another bilinear form ξ T 2 ( V ). Prove that there exists exactly one transformation T so ξ = v T .

Answer & Explanation

Tianna Deleon

Tianna Deleon

Beginner2022-06-28Added 29 answers

By using the fact that ( A B ) T = B T A T (where T denotes a transposed matrix), we get:
[ x ] T [ v t ] [ y ] = ( [ T ] [ x ] ) T [ v ] [ y ] = [ x ] T [ T ] T [ v ] [ y ] = [ x ] T ( [ v ] T [ T ] ) T [ y ]
So, that brings us to:
[ ξ ] T = [ v ] T [ T ]
Notice [ v ] is invertible, as v is non-degenerate, so:
[ T ] = ( [ v ] T ) 1 [ ξ ] T

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