\[\phi\]
from a vector space \[V\]
to a field \[F\]
(often the set of real numbers (\[\mathbb{R}\]
) is called a linear functional if for every \[\mathbf{v}_1, \; \mathbb{R} \in V\]
,\[\phi(\mathbb{v}_1 + \mathbb{v}_2)= \phi (\mathbb{v}_1)_ + \phi (\mathbb{v}_2) \]
(1)\[\phi(c \mathbb{v})= c \phi( \mathbb{v}) \]
(2)where
\[c in F\]
.Every linear functional is a linear transformation.
Examples:
The projection functions
\[\phi(\begin{pmatrix}x_1\\x_2\\ \vdots\\ x_n \end{pmatrix})=x_i\]
\[\phi(x,y)=x-3y\]
.Every linear functional, like every linear transformation, can be represented by a matrix operator. The matrix operator representing the second example above is
\[T=(1 \:-3)\]
and then \[\phi (\begin{pmatrix}x\\y\end{pmatrix})= (1 \; -3) \begin{pmatrix}x\\y\end{pmatrix}=x-3y\]
.