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<p style="margin-top:0;margin-bottom:0">Depends, literature tends to use the definitions interchangeably depending on what aspect is important to subject.</p>
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<div id="divRplyFwdMsg" dir="ltr"><font face="Calibri, sans-serif" style="font-size:11pt" color="#000000"><b>Från:</b> MacOSX-TeX <macosx-tex-bounces@email.esm.psu.edu> för Martin Berggren <martin.berggren@cs.umu.se><br>
<b>Skickat:</b> den 21 augusti 2018 13:59<br>
<b>Till:</b> TeX on Mac OS X Mailing List<br>
<b>Ämne:</b> Re: [OS X TeX] Latex symbol for "define equal"</font>
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<div class="">On 21 Aug 2018, at 13:47, Markus Klyver <<a href="mailto:markusklyver@hotmail.com" class="OWAAutoLink" id="LPlnk315813" previewremoved="true">markusklyver@hotmail.com</a>> wrote:</div>
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<div class="" style="margin-top:0px; margin-bottom:0px">The thing is that you can have several different definitions, all equivalent. Consider the definition "a matrix A \in \mathbb R^{n \times n} is invertiable iff A have a multiplicative inverse"<span class="" style="font-size:12pt">.
It turns out that this is<span class="x_Apple-converted-space"> </span></span>equivalent<span class="" style="font-size:12pt"> to a lot of things, among det(A) !=0, A having full rang, A having n linear</span>independent<span class="" style="font-size:12pt"> eigenvectors,
Ax=0 only having the trivial solution, Ax=b having a solution (which is unique) for every right-hand-side b, etc.</span></div>
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I would save that this is a theorem, not a definition. I think of a definition as a “macro”; that is, giving a short name to a mathematical property. Example: a matrix A is called
<i class="">positive semidefinite</i> when x^T Ax \geq 0 for all vectors x. The point is that you in each instances when the name is used, it can be replaced by its definition. </div>
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<div>Sincerely,</div>
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<div class="">Martin Berggren</div>
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<div class="">Department of Computing Science, </div>
<div class="">UMIT Research Lab</div>
<div class="">Umeå Universitet<br class="">
Campustorget 5, S-901 87 Umeå, Sweden. Ph: +46-70-732 8111<br class="">
<a href="http://www.cs.umu.se/~martinb" class="OWAAutoLink" id="LPlnk217643" previewremoved="true">http://www.cs.umu.se/~martinb</a>, <a href="mailto:Martin.Berggren@cs.umu.se" class="OWAAutoLink" id="LPlnk507351" previewremoved="true">Martin.Berggren@cs.umu.se</a></div>
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