Duality Flashcards
What is a dual space V* of V (vector space over F) and what is their form
Dual space V* of V is:
V* = L(V,F) = alpha V to F | alpha is linear
Elements are called linear functionals
Their form is linear combo of coordinates, E.g
A(x) = A1x1 +…AnXn E.g 167x1 - 3x2 gives A in F^3
What is alpha(v1 + Lv2)
Alpha(v1 + Lv2) = alpha(v1) + Lalpha(v2)
What are properties of V*
Properties of V* are:
Pointwise addition (a1 + a2)(v) = a1(v) + a2(v)
Scalar multiplication (La)(v) = L(a(v))
What is a dual basis and how is it defined
Dual basis is v1,…,vn is a basis of V, v1,…,vn is basis of V
Defined by vi(vj) = delta(ij) = 1 if I = j and 0 if not.
If dual basis element applied to basis vector = 1
What is the sufficiency principle
Sufficiency principle is:
A(v) = 0 iff v = 0 for all A in V*
What is double dual of V
Double dual of V is applying duality to V* to get V**
What ev : V to V**
Ev : = to V** is an isomorphism
Vector space with ev isomorphism is reflexive
What is solution set of E <= V* (system of Linear equations)
Solution set E <= V* is:
Sol E = [ v in V | A(v) = 0 for all A in E) = N (A in E) ker(A) (intersection of kernels) <= V
What is dim sol E
Dim sol E = dim V - dim E
When is A1,…,An in V* a basis
A1,…,An in V* is a basis when N i =1 to n ker(Aj) = 0 (n is dim V)
See example after corollary 5.7
What are properties of solution set (E,F <= V*)
Properties of solution set are:
E <= F implies sol F <= sol E
Sol(E+F) = sol E N sol F
Sol E + sol F <= sol(E N F) with equality if V finite dim
What is annihilator of U <= V*
Annihilator of U <= V is [alpha in V* | alpha (u) = 0 for all u in U]
Ann U or U^0 <= V*
How to know if A1,…,An in V* are a basis (n = dim V*)
To if A1,..,An are a basis of V* is if ker(Ai) = 0
