Abstract Algebra: Theory and Applications · Problem 20.18

Fields — Thomas Judson

Mathematics — Abstract Algebra · dual-spaces

Problem

Dual Spaces. Let

V

and

W

be finite dimensional vector spaces over a field

F

. (a) Show that the set of all linear transformations from

V

into

W

, denoted by

\text{Hom}(V, W)

, is a vector space over

F

, where we define vector addition as follows:

(S + T)(v) = S(v) + T(v)

(\alpha S)(v) = \alpha S(v),

where

S, T \in \text{Hom}(V, W)

\alpha \in F

, and

v \in V

. (b) Let

V

be an

F

-vector space. Define the dual space of

V

to be

V^* = \text{Hom}(V, F)

. Elements in the dual space of

V

are called linear functionals. Let

v_1, \ldots, v_n

be an ordered basis for

V

. If

v = \alpha_1 v_1 + \cdots + \alpha_n v_n

is any vector in

V

, define a linear functional

\phi_i : V \to F

\phi_i(v) = \alpha_i

. Show that the

\phi_i

's form a basis for

V^*

. This basis is called the dual basis of

v_1, \ldots, v_n

(or simply the dual basis if the context makes the meaning clear). (c) Consider the basis

\{(3,1),(2,-2)\}

for

\mathbb{R}^2

. What is the dual basis for

(\mathbb{R}^2)^*

? (d) Let

V

be a vector space of dimension

n

over a field

F

and let

V^{**}

be the dual space of

V^*

. Show that each element

v \in V

gives rise to an element

\lambda_v

V^{**}

and that the map

v \mapsto \lambda_v

is an isomorphism of

V

with

V^{**}