|
|
|
Viewing Version
14
of
'linear transformation'
|
[ view 'linear transformation'
|
back to history
]
| Title of object: |
linear transformation |
| Canonical Name: |
LinearTransformation |
| Type: |
Definition |
| Created on: |
2001-11-07 16:34:10 |
| Modified on: |
2007-02-08 07:58:00 |
| Classification: |
msc:15A04 |
| Synonyms: |
linear transformation=linear map
linear transformation=vector space homomorphism
linear transformation=linear mapping
linear transformation=linear operator |
Revision comment (for changes between this and next version):
Preamble:
\usepackage{amsmath}
\usepackage{mathrsfs}
\def\Ker{\operatorname{Ker}}
\def\Im{\operatorname{Im}}
\def\Hom{\operatorname{Hom}} |
Content:
\PMlinkescapeword{properties}
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}
\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 $\lambda \in F$
\end{itemize}
The set of all linear maps $V \to W$
is denoted by $\Hom_F(V,W)$ or $\mathscr{L}(V,W)$.
\noindent \textbf{Properties:}
\begin{itemize}
\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 The \PMlinkname{kernel}{KernelOfALinearTransformation}
$\Ker(T)=\{v\in V \mid T(v) = 0\}$ is a subspace of $V$.
\item The \PMlinkname{image}{ImageOfALinearTransformation} $\Im(T) = \{T(v) \mid v\in V\}$ is a subspace of $W$.
\item The inverse image $T^{-1}(w)$ is a subspace if and only if $w=0$.
\item A linear transformation is injective if and only if $\Ker(T)=\{0\}$.
\item If $v \in V$ then $T^{-1}T(v) = v + u$ where $u$ is any element of $\Ker(T)$.
\item If $T$ is a surjection and $w\in W$ then $TT^{-1}(w) = w$.
\end{itemize}
\textbf{See also:}
\begin{itemize}
\item Wikipedia, \PMlinkexternal{linear transformation}{http://www.wikipedia.org/wiki/Linear_transformation}
\end{itemize} |
|
|
|
|
|
