Wikipedia: History of Bilinear operator

Difference (from prior major revision) (no other diffs)

Changed: 22,23c22

Examples

Changed: 25,28c24,30

* If a vector space over the real numbers, R, has an inner product, then it is a bilinear operator.
* The application operator, b(f, v) = f(v) is a bilinear operator from V^*xV.
* Let f be a member of V^* and g a member of W^*. Then b(v, w) = f(v)g(w) is a bilinear operator.
* In R², assign to each two vectors the signed area of the parallelogram they define.

One often thinks of a bilinear operator as a generalized "multiplication" which satisfies the [distributive law]?.

* Matrix multiplication is a bilinear map M(m,n) x M(n,p) -> M(m,p).
* If a vector space V over the real numbers R carries an inner product, then the inner product is a bilinear operator V x V -> R.
* If V is a vector space with dual space V*, then the application operator, b(f, v) = f(v) is a bilinear operator from V^*xV to the base field.
* Let V and W be vector spaces over the same base field F. If f is a member of V^* and g a member of W^*, then b(v, w) = f(v)g(w) defines a bilinear operator V x W -> F.
* The cross product in R³ is a bilinear operator R³ x R³ -> R³.

Changed: 30c32

* The operator B: VxW -> X where B(v, w) = 0 is bilinear

* The operator B: VxW -> X where B(v, w) = 0 for all v in V and w in W is bilinear

	Revision 4 . . October 5, 2001 1:30 am by AxelBoldt [expanded a little on examples]
	Revision 3 . . October 4, 2001 3:52 pm by (logged).115.22.xxx [examples]