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