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Version 11 |
| \PMlinkescapeword{properties} |
\PMlinkescapeword{properties} |
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| Let $V$ and $W$ be vector spaces over the same field $F$. A \emph{linear transformation} is a function $T\colon V \to W$ such that: |
Let $V$ and $W$ be vector spaces over the same field $F$. A \emph{linear transformation} is a function $T\colon V \to W$ such that: |
| \begin{itemize} |
\begin{itemize} |
| \item $T(v+w) = T(v)+T(w)$ for all $v,w \in V$ |
\item $T(v+w) = T(v)+T(w)$ for all $v,w \in V$ |
| \item $T(\lambda v) = \lambda T(v)$ for all $v\in V$, and $\lambda \in F$ |
\item $T(\lambda v) = \lambda T(v)$ for all $v\in V$, and $\lambda \in F$ |
| \end{itemize} |
\end{itemize} |
| and the set of all linear maps $V \to W$ is denoted as $\mathscr{L}(V,W)$. |
and the set of all linear maps $V \to W$ is denoted as $\mathscr{L}(V,W)$. |
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| \noindent \textbf{properties:} |
\noindent \textbf{properties:} |
| \begin{itemize} |
\begin{itemize} |
| \item $T(0) = 0$. |
\item $T(0) = 0$. |
| \item If $G\colon W\to U$ is a linear transformations then $G\circ T\colon V\to U$ is also a linear transformation. |
\item If $G\colon W\to U$ is a linear transformations then $G\circ T\colon V\to U$ is also a linear transformation. |
| \item The \PMlinkname{kernel}{KernelOfALinearTransformation} |
\item The \PMlinkname{kernel}{KernelOfALinearTransformation} |
| $\operatorname{ker}(T)=\{v\in V \mid T(v) = 0\}$ is a subspace of $V$. |
$\operatorname{ker}(T)=\{v\in V \mid T(v) = 0\}$ is a subspace of $V$. |
| \item The \PMlinkname{image}{ImageOfALinearTransformation} $\operatorname{Im}(T) = \{T(v) \mid v\in V\}$ is a subspace of $W$. |
\item The \PMlinkname{image}{ImageOfALinearTransformation} $\operatorname{Im}(T) = \{T(v) \mid v\in V\}$ is a subspace of $W$. |
| \item The inverse image $T^{-1}(w)$ is a subspace if and only if $w=0$. |
\item The inverse image $T^{-1}(w)$ is a subspace if and only if $w=0$. |
| \item A linear transformation is injective if and only if $\operatorname{ker}(T)=\{0\}$. |
\item A linear transformation is injective if and only if $\operatorname{ker}(T)=\{0\}$. |
| \item If $v \in V$ then $T^{-1}T(v) = v + u$ where $u$ is any element of $\operatorname{ker}(T)$. |
\item If $v \in V$ then $T^{-1}T(v) = v + u$ where $u$ is any element of $\operatorname{ker}(T)$. |
| \item If $T$ is a surjection and $w\in W$ then $TT^{-1}(w) = w$. |
\item If $T$ is a surjection and $w\in W$ then $TT^{-1}(w) = w$. |
| \end{itemize} |
\end{itemize} |
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| \textbf{see also:} |
\textbf{see also:} |
| \begin{itemize} |
\begin{itemize} |
| \item Wikipedia, \PMlinkexternal{linear transformation}{http://www.wikipedia.org/wiki/Linear_transformation} |
\item Wikipedia, \PMlinkexternal{linear transformation}{http://www.wikipedia.org/wiki/Linear_transformation} |
| \end{itemize} |
\end{itemize} |
