Linear Functional

Any Linear Map that takes in a vector space and outputs a scalar in the same field.

Formal Definition

• With $V$ as a Vector Space over Field $\mathbb{F}$
• Linear Map $f\in \mathcal{L}(V,\mathbb{F})$ are called linear functionals
• Linear functional under a vector $\alpha$ are denoted as: $f_{\alpha }(x)=a\cdot x$

List

• Matrix Trace

Example Proof

1. With $V$ as the vector space of Polynomials
2. Let $t\in \mathbb{F}$
3. Show that $L_t(P)=P(t),\forall P\in V$ defines a linear functional $L\in \mathcal{L}(V<\mathbb{F})$

Soln

1. As $P(t)\in \mathbb{F},\forall t\in \mathbb{F}$
2. $L_t(P):V\to \mathbb{F}$
3. Let $c\in \mathbb{F}$
4. let $f,g\in V$
5. Then, $L_t(cf+g)=(cf+g)(t)=cf(t)+g(t)=c{L}_t(f)+L_t(g)$
6. Thus, this is a linear functional