Definition of complex numbers (n∈C)
C = { a+bi : a,b ∈ R } |
addition in C (a+bi) + (c+di) = (a+c) + (b+d)i |
multiplication in C (a+bi)(c+di) = (ac-bd) + (ad+bc)i |
Properties of Fields
commutativity a+b = b+a ∧ ab = ba ∀a,b ∈ F |
associativity (a+b)+c = a+(b+c) ∧ (ab)c = a(bc) ∀a,b,c ∈ F |
identities c+0 = c ∧ 1c = c ∀c ∈ F |
additive inverse ∀a ∈ F ∃b ∈ F : a+b = 0 |
multiplicative inverse a ∈ F, a ≠ 0, ∃b ∈ F : a+b = 0 |
distributive property c(a+b) = ca + cb ∀a,b,c ∈ F |
Definition of list L with length n
n∈N, L : { 1,2,…,n } → Elements |
L = (x1,...,xn) |
Definition of Fn
Fn = { (x1,...,xn) : xj ∈ F for j = 1,...,n} |
addition in Fn (x1,...,xn) + (y1,...,yn) = (x1+y1,...,xn+yn) |
zero vector 0 = (0,...,0) |
additive inverse For x ∈ Fn, -x ∈ Fn = (-x1,...,-xn) |
scalar multiplication in Fn c ∈ F, c(x1,...,xn) = (cx1,...,cxn) |
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Definition of addition and scalar multiplication
addition on a set V + : (u,v ∈ V) → u +v |
scalar multiplication on a set V * : (c ∈ F, v ∈ V) → cv |
Defintion of Vector Space
Vector Space = V + (+) + (*) |
commutativity u+v = v+u |
associativity (u+v)+w = u+(v+w) ∧ (ab)v = a(bv) |
additive identity ∃0 ∈ V : v+0 = v |
additive inverse ∀v ∈ V, ∃w ∈ V : v+w = 0 |
multiplicative identity 1v = v |
distributive property a(v+w) = av + aw ∀a ∈ F |
v ∈ V = vector ∨ point |
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