Dual Spaces and Dual Maps

Dual Spaces

Definition 100: Linear Functionals and Dual Spaces

Let $𝑉$ be a vector space over a field $𝐹$ . A linear functional on $𝑉$ is a linear map from $𝑉$ to $𝐹$ . The set of all linear functionals on $𝑉$ is called the dual space of $𝑉$ , denoted $𝑉^{*}$ :

𝑉^{*} = ℒ (𝑉, 𝐹) .

Example.

Let $𝑉 = ℝ^{2}$ . Then the dual space of $𝑉$ is $𝑉^{*} = ℒ (ℝ^{2}, ℝ)$ , which can be described as:

𝑉^{*} = {𝑓 : ℝ^{2} \to ℝ | 𝑓 is linear} = {𝑓 : ℝ^{2} \to ℝ | 𝑓 (𝑥, 𝑦) = 𝑎 𝑥 + 𝑏 𝑦 for some 𝑎, 𝑏 \in ℝ} .

Theorem 82: Dual Basis

Suppose $𝑉$ is a finite-dimensional vector space over a field $𝐹$ and $ℬ = {𝐯_{1}, 𝐯_{2}, \dots, 𝐯_{𝑛}}$ is a basis for $𝑉$ . Then there exists a unique set of linear functionals $ℬ^{*} = {𝜑_{1}, 𝜑_{2}, \dots, 𝜑_{𝑛}}$ , called the dual basis of $ℬ$ , such that:

{\begin{matrix} 𝜑_{𝑖} (𝐯_{𝑖}) = 1 for all 𝑖 \\ 𝜑_{𝑖} (𝐯_{𝑗}) = 0 for all 𝑖 \neq 𝑗 \end{matrix}

where $ℬ^{*}$ is a basis for the dual space $𝑉^{*}$ .

Corollary 43: Dimension of Dual Spaces

Let $𝑉$ be a finite-dimensional vector space over a field $𝐹$ . Then $\dim (𝑉^{*}) = \dim (𝑉)$ .

Corollary 44: Isomorphism between a Finite-Dimensional Vector Space and its Dual

Let $𝑉$ be a finite-dimensional vector space over a field $𝐹$ . Then $𝑉 ≅ 𝑉^{*}$ .

Theorem 83: Dual Basis Theorem

Let $ℬ = {𝐛_{1}, \dots, 𝐛_{𝑛}}$ be an ordered basis for a finite-dimensional vector space $𝑉$ , and suppose $ℬ^{*} = {𝜑_{1}, \dots, 𝜑_{𝑛}}$ is the dual basis of $ℬ$ . Then for any vector $𝐯 \in 𝑉$ , the $𝜑_{𝑖} (𝐯)$ evaluates to the $𝑖$ ^th coordinate of $𝐯$ with respect to the basis $ℬ$ . That is:

𝐯 = \sum_{𝑖 = 1}^{𝑛} 𝜑_{𝑖} (𝐯) 𝐛_{𝑖} = 𝜑_{1} (𝐯) 𝐛_{1} + 𝜑_{2} (𝐯) 𝐛_{2} + \dots + 𝜑_{𝑛} (𝐯) 𝐛_{𝑛} .

Definition 101: Double Dual

Let $𝑉$ be a vector space over a field $𝐹$ . The double dual of $𝑉$ , denoted $𝑉^{* *}$ , is the dual space of the dual space of $𝑉$ :

𝑉^{* *} = {(𝑉^{*})}^{*} = ℒ (𝑉^{*}, 𝐹) = ℒ (ℒ (𝑉, 𝐹), 𝐹) .

Theorem 84: Natural Isomorphism

Let $𝑉$ be a finite-dimensional vector space over a field $𝐹$ . Define the linear map $𝜂 : 𝑉 \to 𝑉^{* *}$ by:

𝜂 (𝐯) = (𝜑 \mapsto 𝜑 (𝐯)) for all 𝐯 \in 𝑉 .

It holds that $𝜂$ is an isomorphism, so $𝑉 ≅ 𝑉^{* *}$ .

Dual Maps

Definition 102: Dual Map

Let $𝑇 : 𝑉 \to 𝑊$ be a linear map. The dual map of $𝑇$ , denoted $𝑇^{*}$ , is the map $𝑇^{*} : 𝑊^{*} \to 𝑉^{*}$ defined for all $𝜑 \in 𝑊^{*}$ by:

𝑇^{*} (𝜑) = 𝜑 \circ 𝑇 .

Proposition 24: Dual Map is Linear

Let $𝑇 : 𝑉 \to 𝑊$ be a linear map. Then its dual map $𝑇^{*} : 𝑊^{*} \to 𝑉^{*}$ is also a linear map.