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bilinear map fixed while letting the first entry vary, yielding , the result is a linear operator, and similarly for when we hold the first entry fixed.

Such a map satisfies the following properties.

* For any ,
* The map is additive in both components: if and then and

If and we have 1=''B''(''v'', ''w'') = ''B''(''w'', ''v'') for all then we say that B is symmetric. If X is the base field F, then the map is called a bilinear form,

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