1. Dual Space Let V be a vector space over a field F: A linear functional on V is a linear map f : V ! F: The set of all linear functions on V is denoted by V ∗: By definition, V ∗ = L(V; F ): The set L(V; F ) is also a vector space over F: We assume that V is an n dimensional vector space over F: Let β = fv1; ··· ; vng be a basis for V: For each v 2 V; there exist unique numbers a1; ··· ; an 2 F such that v = a1v1 + ··· + anvn: For each 1 ≤ i ≤ n; we define fi(v) = ai: Then fi : V ! F defines a linear functional on V for each 1 ≤ i ≤ n: ∗ Definition 1.1. The ordered set β = ff1; ··· ; fng is called the set of coordinate functions on V with respect to the basis β: It follows from the definition that fi(vj) = δij for 1 ≤ i; j ≤ n: Theorem 1.1. The set β∗ forms an ordered basis for V ∗ such that for any f 2 V ∗; n X (1.1) f = f(vi)fi: i=1 ∗ Proof. Let us prove that β is linearly independent over F: Suppose a1f1 + ··· + anfn = 0: By fi(vj) = δij; for each 1 ≤ j ≤ n; one has (a1f1 + ··· + anfn)(vj) = a1f1(vj) + ··· + anfn(vj) = aj = 0: Now let us prove that β∗ and (1.1) at the same time. Pn ∗ Let f be given. Define g : V ! F by g(v) = i=1 f(vi)fi(v): Then g 2 V : Furthermore, for each 1 ≤ j ≤ n; n n X X g(vj) = f(vi)fi(vj) = f(vi)δij = f(vj): i=1 i=1 We see that the two linear functionals f and g coincide on β: Sine f; g are linear, f = g on ∗ V: We find that f 2 span β and f has the representation of the form (1.1). Definition 1.2. The ordered basis β∗ is called the dual basis to β: ∗ Corollary 1.1. dimF V = dimF V : ∗∗ ∗ ∗ Theorem 1.2. Let V be the dual space to V : For each v 2 V; we define vb : V ! F by sending f to f(v): Then (1) vb : V ! F is a linear map; ∗∗ (2) the function ' : V ! V sending v to vb is a linear isomorphism. Proof. We leave it to the reader to check that vb is a linear map. Let us show that ker ' = f0g: Let v 2 ker ': Then '(v) = 0: Hence vb(f) = 0 for any ∗ f 2 V : Write v = a1v1 + ··· + anvn for some a1; ··· ; an 2 F: Since f(v) = 0 for all v 2 V; fi(v) = 0 for all 1 ≤ i ≤ n which implies that ai = fi(v) = 0 for all 1 ≤ i ≤ n: We see that v = 0: Since V ∗∗ is the dual basis to V ∗ and V ∗ is the dual basis to V; by Corollary 1.1, ∗∗ ∗ dimF V = dimF V = dimF V: ∗∗ Since ' : V ! V is a linear monomorphism (linear and injective) with dimF V = ∗∗ dimF V ;' is a linear isomorphism. 1.