• Linear Functionals and Dual Spaces
  • Linear Functionals
  • Examples
  • Properties of Linear Functionals
  • Significance

📐 Today’s Lesson: Linear Functionals and Dual Spaces From: Advanced Linear Algebra (Roman) — Section 20

§Linear Functionals and Dual Spaces

Linear transformations from a vector space V to the base field F are extremely important objects that capture the notion of “measuring” vectors. These special transformations form a vector space of their own, called the dual space.

§Linear Functionals

Definition (Linear Functional): Let V be a vector space over F. A linear transformation f ∈ L(V, F), whose values lie in the base field F, is called a linear functional (or simply functional) on V.

Definition (Dual Space): The vector space of all linear functionals on V is denoted by V^* and is called the algebraic dual space of V.

§Examples

Example 1 (Evaluation): The map f: F[x] → F defined by f(p(x)) = p(0) is a linear functional, known as evaluation at 0.

Example 2 (Integration): Let C[a,b] denote the vector space of all continuous functions on [a,b] ⊆ ℝ. The map f: C[a,b] → ℝ defined by

f(α(x)) = ∫ₐᵇ α(x) dx

is a linear functional on C[a,b].

§Properties of Linear Functionals

For any f ∈ V^*, the rank plus nullity theorem gives:

((f)) + (im(f)) = (V)

Since im(f) ⊆ F, we have either:

• im(f) = 0, in which case f is the zero linear functional, or • im(f) = F, in which case f is surjective

In other words, a nonzero linear functional is surjective. Moreover, if f ≠ 0:

codim((f)) = (V/(f)) = 1

And if (V) Theorem:

• For any nonzero vector v ∈ V, there exists a linear functional f ∈ V^* for which f(v) ≠ 0. • A vector v ∈ V is zero if and only if f(v) = 0 for all f ∈ V^. • Let f ∈ V^. If f(x) ≠ 0 then V = ⟨ x ⟩ ⊕ (f). • Two nonzero linear functionals f, g ∈ V^* have the same kernel if and only if there is a nonzero scalar λ such that f = λ g.

§Significance

Thus, in dimensional terms, the kernel of a linear functional is a very “large” subspace of the domain V. Hyperplanes (subspaces of codimension 1) are precisely the kernels of nonzero linear functionals.

Key Insight: Linear functionals provide a way to “coordinatize” vectors without choosing a basis. They form the foundation for the theory of dual spaces and play a crucial role in functional analysis, differential geometry, and physics.

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