179 Linear Functional
A linear mapping from a vector space to scalars that is used to define the dual space.
definition (Linear Functional) A linear transformation from a vector space \(V\) to its associated field of scalars \(K\), The mapping \(f: V \rightarrow K\) must satisfy the following linearity condition:
- \(f(ax + by) = af(x) + bf(y)\)
where
- \(x, y \in V\) are vectors
- \(a, b \in K\) are scalars
- \(V\) is a vector space
- \(K\) is the field of scalars.
179.1 Elementary Example
179.1.1 Simple
A linear functional maps each vector to a scalar.
\[ f : V \rightarrow \mathbb{R} \]
\[ V = \{ e_{1},\ e_{2},\ e_{3} \} \]
\[ f(e_{1}) = 1,\quad f(e_{2}) = -2,\quad f(e_{3}) = 0 \]
where
- \(f\) is the linear functional.
- \(V\) is the set of basis vectors used as inputs.
179.1.2 General
On \(\mathbb{R}^{3}\), \(f\) is given by a row of components \(\omega_{i}\) via the dot product with those components.
\[ f(v) = \omega_{1} v^{1} + \omega_{2} v^{2} + \omega_{3} v^{3} \]
\[ (\omega_{1},\omega_{2},\omega_{3}) = (1,-2,0) \]
where
- \(\omega_{i}\) are the components of \(f\) in the dual basis.
- \(v^{i}\) are the components of \(v\).
- Alternating Function
- Alternating k-linear function
- Boundary Operation
- Bundle
- Cartesian Product
- Constant Metric Tensor
- Contravariant Rank
- Covariant Rank
- Covariant, Contravariant, and Coordinate Systems
- Differential Forms
- Differential k-Form
- Edge of a Domain
- Exterior Algebra and Exterior Derivatives
- Forms
- Functions
- Inverse Metric
- k-Covector
- k-Covector Field
- k-Fold Product
- k-form
- k-Tensor
- Lie
- Lie Algebra
- Line Element
- Linear Functional
- Lorentzian Manifold
- Metric Tensor
- Multilinear Function
- Nondegenerate
- Nondegenerate Bilinear Form
- One-Form
- Smooth
- Smooth Assignment
- Smooth Mapping
- Stoke’s Theorem and the Fundamental Theorem of Calculus
- Symmetric Array
- Tangent Space
- Tensor Field
- Tensors
- Two-Form
- Type-(0,2) Tensor Field
- Type-(q,r) Tensor