Linear Functional - Definition and Examples

A mapping  
\[\phi\]
  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  
\[\phi (\begin{pmatrix}x\\y\end{pmatrix})= (1 \; -3) \begin{pmatrix}x\\y\end{pmatrix}=x-3y\]
.

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