DG

Vector space

Vector - [L. vehere to carry] 3. A carrier of disease.

Definition: Vectors are things that can be added and scaled.

History:

Vectors were first used by Bernhard Placidus Johann Nepomuk Bolzano in Betrachtungen über einige Gegenstände der Elementargeometrie (1804)
In 1832 Hermann Grassmann derived the vector forms of laws of mechanics. He went on to define exterior product and introduced the notions of linear subspace and linear independence.
William Rowan Hamilton interpreted complex numbers as pairs of real numbers in 1837, considered n-tuples of real numbers in 1841 and through his invention and work on quaternions went on to give definitions of dot and cross products in 3-dimensional space.
Abstract vector spaces (linear spaces) were introduced by Giuseppe Peano in Calcolo geometrico secondo l'Ausdehnungslehre di H. Grassmann preceduto dalle operazioni della logica deduttiva (1888)
Vectors were popularized by Edwin Bidwell Wilson in Vector Analysis (1901) after the Yale lectures of Josiah Willard Gibbs in 1899.

Definition: A vector space (a.k.a. linear space) V is a set with addition (+) and multiplication by scalars (·) a field K such that $\forall$ u, v, w $\in$ V; r, s $\in$ K

u+(v+w) = (u+v)+w
u+v = v+u
r·(u+v) = r·u+r·v
(r+s)·u = r·u+s·u
r·(s·u) = (rs)·u
1·u = u, 0·u = 0

Note: A vector space is a special case of a module.

The combined operations give linear combinations:
$\sum$ _{k = 1}^m r_k·u_k = r₁·u₁+...+r_m·u_m, where u_k $\in$ V, r_k $\in$ K.

Example 1: Real n-dimensional Euclidean space Eⁿ (e.g. the plane E²) can be made into a real vector space by a choice of origin.

Pick any point p of Eⁿ. Given two points u, v $\in$ Eⁿ, draw segments connecting them to the origin p. Complete the parallelogram. The sum u+v is defined to be the remaining vertex of the parallelogram.

To scale a point u by a positive real number r draw a ray from p through u and define v = r·u to be a point on the ray such that the ratio of the distances from p to v and p to u is r. To scale by -1 reflect u about p.

Note: This example illustrates the geometric aspect of Euclidean vectors (drawn as arrows) as models for phenomena having both direction and magnitude, e.g. momentum.

Example 2: Let Rⁿ be the set of all sequences of real numbers of length n (ordered n-tuples). Placed horizontally these sequences are known as row vectors. Often they are placed vertically as column vectors. Define addition and scaling coordinate-wise:

(a₁,...a_n)+(b₁,...b_n) = (a₁+...a_n+b_n)
c(a₁,...a_n) = (ca₁,...ca_n)

Note that allowing n = $\infty$ does not change things.

Example 2: Let $\sC$ (R) be the set of all continuous R -> R. Define addition and scaling as follows:

(f₁+f₂)(x) = f₁(x)+f₂(x)
(cf)(x) = cf(x)

Note that addition and scaling preserve continuity. Can we do the same with differentiable functions?

Exercise: Verify that the sets with operations as in the above examples satisfy the axioms of a vector space over R.

Last updated: Jun 17 18:53 / Last fetched: Tue Dec 2 10:03:50 CST 2008