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 = {v1, · · · , vn} be a basis for V. For each v ∈ V, there exist unique numbers a1, · · · , an ∈ 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∗ = {f1, · · · , fn} 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 ∈ V∗,
(1.1) f =
n
X
i=1
f (vi)fi.
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.
Let f be given. Define g : V → F by g(v) =Pn
i=1f (vi)fi(v). Then g ∈ V∗. Furthermore, for each 1 ≤ j ≤ n,
g(vj) =
n
X
i=1
f (vi)fi(vj) =
n
X
i=1
f (vi)δij = f (vj).
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 ∈ 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. dimFV = dimFV∗.
Theorem 1.2. Let V∗∗ be the dual space to V∗. For each v ∈ V, we define bv : V∗ → F by sending f to f (v). Then
(1) bv : V → F is a linear map;
(2) the function ϕ : V → V∗∗ sending v tov is a linear isomorphism.b Proof. We leave it to the reader to check thatv is a linear map.b
Let us show that ker ϕ = {0}. Let v ∈ ker ϕ. Then ϕ(v) = 0. Hence bv(f ) = 0 for any f ∈ V∗. Write v = a1v1+ · · · + anvn for some a1, · · · , an∈ F. Since f (v) = 0 for all v ∈ 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,
dimFV∗∗= dimFV∗ = dimF V.
Since ϕ : V → V∗∗ is a linear monomorphism (linear and injective) with dimFV =
dimF V∗∗, ϕ is a linear isomorphism.
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