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Linear mappings.

Definition 6.1.1   Let V and W be two vector spaces. A mapping

\begin{displaymath}\phi: V \longrightarrow W
\end{displaymath}

is a linear mapping or a linear operator if it verifies the following properties:
1.
$\forall \overrightarrow{u} , \overrightarrow{v}\in V, \;
\phi ( \overrightarro...
... + \overrightarrow{v} )=\phi (\overrightarrow{u} ) + \phi (\overrightarrow{v} )$.
2.
$\forall \overrightarrow{u} , \forall \alpha \in \mathbb{K} ,
\phi (\alpha \overrightarrow{u} )= \alpha \; \phi (\overrightarrow{u} )$.

Proposition 6.1.2   Let V and W be two vector spaces. A mapping

\begin{displaymath}\phi: V \longrightarrow W
\end{displaymath}

is a linear mapping if, and only if, it verifies the following property:

\begin{displaymath}\forall \overrightarrow{u} , \overrightarrow{v}\in V, \;
\fo...
...i (\overrightarrow{u} ) + \beta \; \phi (\overrightarrow{v} ).
\end{displaymath}

Proposition 6.1.3   Let U,V,W be three vector spaces with respective dimension m,n,p, in which bases are given. Let $\phi : U \longrightarrow V$ and $\psi : V \longrightarrow W$ be two linear mappings. Then $ \psi o \phi : U \longrightarrow W$ is a linear mapping.

Remark 6.1.4       

If $A=(a_{ij})_{\begin{matrix}1 \leq i \leq m \\ 1 \leq j \leq n \end{matrix}}$ is the matrix of $\phi$ and $B=(b_{ij})_{\begin{matrix}1 \leq i \leq n \\ 1 \leq j \leq p \end{matrix}}$ w.r.t. the given bases, then the matrix of $ \psi o \phi$ is

\begin{displaymath}C=(c_{ij})_{\begin{matrix}1 \leq i \leq m \\ 1 \leq j \leq p \end{matrix}}
\end{displaymath}

where

\begin{displaymath}c_{ij}=\underset{k=1}{\overset{n}{\sum}} a_{ik}b_{kj}.
\end{displaymath}

Proposition 6.1.5   If U and V are two $\bbb{K}-$vector spaces, the set $\mathcal{L}(U,V)$ of all the linear mappings from U to V is a $\bbb{K}-$vector space.

Remark 6.1.6   Let $\phi : \; U \longrightarrow V$ be a linear application.


next up previous contents
Next: Kernel and Image of Up: Linear mappings. Previous: Linear mappings.
Noah Dana-Picard
2001-02-26