[Axler] Duality

Linear Algebra: Duality (Summary)

Based on Linear Algebra Done Right by Sheldon Axler

A summary of the Dual Space, Dual Basis, Dual Maps, and Annihilators.

---

1. Dual Space ($V'$)

Definition

The dual space of a vector space $V$ over a field $F$, denoted as $V'$, is the set of all linear functionals on $V$.

• A linear functional is a linear map $\varphi : V \to F$.

Properties

• Vector Space Structure: $V'$ is itself a vector space under pointwise addition and scalar multiplication.
• Dimension: If $V$ is finite-dimensional ($\dim V = n$), then $\dim V' = n$.

---

2. Dual Basis

Let $v_1, \dots, v_n$ be a basis of $V$. The dual basis $\varphi_1, \dots, \varphi_n$ of $V'$ is defined by:

$$ \varphi_j(v_k) = \delta_{jk} $$

Where $\delta_{jk}$ is the Kronecker delta (1 if $j=k$, 0 if $j \ne k$).

Theorem

The list $\varphi_1, \dots, \varphi_n$ is a basis for $V'$.

Proof Sketch:
1. Linear Independence: Suppose $\sum c_i \varphi_i = 0$. Applying this to basis vector $v_k$ yields $c_k = 0$ for all $k$.
2. Spanning: Any functional $\psi \in V'$ can be written as: $$ \psi = \psi(v_1)\varphi_1 + \dots + \psi(v_n)\varphi_n $$

---

3. Dual Map ($T'$)

Definition

For a linear map $T : V \to W$, the dual map $T' : W' \to V'$ is defined by the "pullback" operation:

$$ (T'(\psi))(v) = \psi(T(v)) $$

for all $\psi \in W'$ and $v \in V$.

Properties

• Linearity: $T'$ is a linear map.
• Composition: $(S \circ T)' = T' \circ S'$. (Note the order reversal).
• Invertibility: If $T$ is invertible, then $(T')^{-1} = (T^{-1})'$.
• Matrix Representation: For finite-dimensional spaces, the matrix of $T'$ (relative to dual bases) is the transpose of the matrix of $T$. $$ \mathcal{M}(T') = (\mathcal{M}(T))^t $$

---

4. Double Dual ($V''$)

The double dual $V''$ is the dual space of $V'$. It consists of linear functionals on $V'$.

Natural Isomorphism

If $V$ is finite-dimensional, there is a natural isomorphism $\iota : V \to V''$ defined by evaluation:

$$ \iota(v)(\varphi) = \varphi(v) $$

---

5. Annihilator ($U^0$)

For a subset $U \subseteq V$, the annihilator $U^0 \subseteq V'$ is the set of functionals that map every element of $U$ to 0:

$$ U^0 = \{ \varphi \in V' \mid \varphi(u) = 0 \text{ for all } u \in U \} $$

Key Properties

• $U^0$ is a subspace of $V'$.
• Dimension Formula: If $U$ is a subspace of finite-dimensional $V$: $$ \dim U + \dim U^0 = \dim V $$
• Double Annihilator: $(U^0)^0 = U$ (under the natural isomorphism).

---

6. Examples

Example 1: Polynomial Space ($P_m(\mathbb{R})$)

For basis $1, x, x^2, \dots, x^m$, the dual basis consists of functionals $\varphi_j$ defined by derivatives at 0:

$$ \varphi_j(p) = \frac{p^{(j)}(0)}{j!} $$

(This extracts the coefficient of $x^j$ in the polynomial).

Example 2: Euclidean Space ($\mathbb{R}^n$)

With standard basis $e_1, \dots, e_n$, the dual basis functionals are the coordinate projection functions:

$$ \varphi_j(x_1, \dots, x_n) = x_j $$

---

7. Key Theorems Summary

• Dual Basis Existence: Every basis of $V$ induces a unique dual basis in $V'$.
• Isomorphism: $V \cong V'$ (non-canonical) but $V \cong V''$ (canonical/natural).
• Fundamental Theorem of Linear Maps (Dual Version):
  • $\null(T') = (\range T)^0$
  • $\range(T') = (\null T)^0$
  • $\dim(\range T) = \dim(\range T')$ (Column rank = Row rank)