linear functional

Topics

• linear algebra
• functional analysis

Linear functional maps vectors to scalars while preserving linear structure. For any vector space $V$ over field $K$, linear functional $\varphi$ satisfies:

$$\varphi (\alpha \mathbf{u}+\beta \mathbf{v})=\alpha \varphi (\mathbf{u})+\beta \varphi (\mathbf{v})$$

for all vectors $\mathbf{u},\mathbf{v}\in V$ and scalars $\alpha ,\beta \in K$.

Key properties

• Special case of linear transformation where output space is scalar field $K$
• Real-valued linear functionals map vectors to $\mathbb{R}$

Examples

• Projection: Selecting $i^{th}$ component from vector $\mathbf{v}=(v_1,...,v_n)$
• Integration: Maps continuous function to its definite integral value
• The trace of square matrix (sum of diagonal elements)
• The dot product with fixed vector (when $V$ has inner product)

Warning

• The determinant appears linear but fails additivity condition
• Derivative operator is linear but outputs functions, not scalars

Related

• riesz representation theorem