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Bilinear form
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{{Short description|Scalar-valued bilinear function}} In [[mathematics]], a '''bilinear form''' is a [[bilinear map]] {{math|''V'' Γ ''V'' β ''K''}} on a [[vector space]] {{mvar|V}} (the elements of which are called ''[[Vector (mathematics)|vectors]]'') over a [[Field (mathematics)|field]] ''K'' (the elements of which are called ''[[scalar (mathematics)|scalar]]s''). In other words, a bilinear form is a function {{math|''B'' : ''V'' Γ ''V'' β ''K''}} that is [[linear map|linear]] in each argument separately: * {{math|1=''B''('''u''' + '''v''', '''w''') = ''B''('''u''', '''w''') + ''B''('''v''', '''w''')}} {{spaces|3}} and {{spaces|3}} {{math|1=''B''(''Ξ»'''''u''', '''v''') = ''Ξ»B''('''u''', '''v''')}} * {{math|1=''B''('''u''', '''v''' + '''w''') = ''B''('''u''', '''v''') + ''B''('''u''', '''w''')}} {{spaces|3}} and {{spaces|3}} {{math|1=''B''('''u''', ''Ξ»'''''v''') = ''Ξ»B''('''u''', '''v''')}} The [[dot product]] on <math>\R^n</math> is an example of a bilinear form which is also an [[Inner product space|inner product]].<ref>{{Cite web| date=2021-01-16| title=Chapter 3. Bilinear forms β Lecture notes for MA1212| url=https://www.maths.tcd.ie/~pete/ma1212/chapter3.pdf}}</ref> An example of a bilinear form that is not an inner product would be the [[four-vector]] product. The definition of a bilinear form can be extended to include [[module (mathematics)|modules]] over a [[Ring (mathematics)|ring]], with [[linear map]]s replaced by [[module homomorphism]]s. When {{mvar|K}} is the field of [[complex number]]s {{math|'''C'''}}, one is often more interested in [[sesquilinear form]]s, which are similar to bilinear forms but are [[conjugate linear]] in one argument.
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