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3
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'linear isomorphism'
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| Title of object: |
linear isomorphism |
| Canonical Name: |
LinearIsomorphism |
| Type: |
Definition |
| Created on: |
2004-09-17 13:23:34 |
| Modified on: |
2004-10-17 10:54:29 |
| Classification: |
msc:15A04 |
| Synonyms: |
linear isomorphism=invertible linear map
linear isomorphism=bijective linear map
linear isomorphism=non-singular linear map |
Revision comment (for changes between this and next version):
| Changes for correction #14631 ('U, V... and W?'). |
Preamble:
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\usepackage{amssymb}
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%\usepackage{psfrag}
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\newcommand{\sR}[0]{\mathbb{R}}
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\newcommand{\sN}[0]{\mathbb{N}}
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\newcommand{\Z}{\mathbbmss{Z}}
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Content:
\begin{defn}
Suppose $V$ and $W$ are vector spaces and $L\colon U\to V$ is a linear map.
Then $L$ is a \emph{linear isomorphism} if $L$ is bijective.
\end{defn}
\subsubsection*{Properties}
\begin{enumerate}
\item Compositions and of linear isomorphisms is a linear isomorphism.
\item The inverse of a linear isomorphisms is a linear isomorphism.
\item If either $U$ or $V$ if finite dimensional, then $\dim V=\dim W$.
(This is a consequence of the rank-nullity theorem.)
\end{enumerate} |
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