History of Vector calculus

Difference (from prior major revision) (minor diff, author diff)

Changed: 5,29c5

Vectors live in a space called a "vector space" over a given field? that overloads two operations (vector + vector and scalar * vector) to follow eight rules.
Given vectors u, v, and w, and scalars a and b:

*0 + v = v
*v + w = w + v
*(u + v) + w = u + (v + w)
*0 * v = 0
*1 * v = v
*(a + b) * v = a * v + b * v
*a * (v + w) = a * v + a * w
* (help!)

Examples of vector spaces:
* F^n, over F
** R^n, over R (the real numbers)
** The finite field GF(p^n), over GF(p)
* C (complex numbers), over R
* Given a field F and a vector space V over F, the set of functions F -> V, over F
* R, over Q (the rational numbers).

Conjecture: If A is a vector space over B, and B is a vector space over C, A is a vector space over C.
(Proof? Disproof?)

Sources

*http://everything2.com/?node=vector+space

Vectors live in a space called a "vector space" over a given field.

	Revision 5 . . (edit) October 21, 2001 7:28 pm by Ap
	Revision 4 . . (edit) October 21, 2001 7:27 pm by Ap
	Revision 3 . . October 21, 2001 8:17 am by Damian Yerrick [oops... there's already an article about vector spaces; moving my stuff there]
	Revision 2 . . (edit) October 21, 2001 8:09 am by Damian Yerrick [removed redundant "over F" in "Given a field..."]
	Revision 1 . . October 21, 2001 8:06 am by Damian Yerrick [...to here, with some of the vector space rules]