rbeezer committed 33d3842

Sage CNIP, removed obsolete discussion (Michael DuBois)

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 Edit: Minor edits in Sections WILA, SSLE, RREF, TSS, HSE, NM
 Edit: Minor edits in Section VO, LC, SS, LI, LDS, O
 Edit: Extended slightly the conclusion of Theorem HSC
+Edit: Sage CNIP, removed obsolete discussion (Michael DuBois)
 Change: Theorem ISRN demoted to Exercise TSS.T11
 Change: Prefer "pivot columns" over "leading 1", Chapter SLE
 Typo: C^n in statement of Theorem SLSLC


 The term <q>inner product</q> means slightly different things to different people.  For some, it is the <q>dot product</q> that you may have seen in a calculus or physics course.  Our inner product could be called the <q>Hermitian inner product</q> to emphasize the use of vectors over the complex numbers and conjugating some of the entries.  So Sage has a <code>.dot_product()</code>, <code>.inner_product()</code>, and <code>.hermitian_inner_product()</code> <mdash /> we want to use the last one.<br /><br />
-Furthermore, Sage defines the Hermitian inner product by conjugating entries from the <em>first</em> vector, rather than the <em>second</em> vector as defined in the text.  This is not as big a problem as it might seem.  First, <acroref type="theorem" acro="IPAC" />, tells us that we can counteract a difference in order by just taking the conjugate, so you can translate between Sage and the text by taking conjugates of results achieved from a Hermitian inner product.  Second, we will mostly be interested in when a Hermitian inner product is zero, which is its own conjugate, so no adjustment is required.  Third, most theorems are true as stated with either definition, although there are exceptions, like <acroref type="theorem" acro="IPSM" />.<br /><br />
 From now on, when we mention an inner product in the context of using Sage, we will mean <code>.hermitian_inner_product()</code>.  We will redo the first part of <acroref type="example" acro="CSIP" />.  Notice that the syntax is a bit asymmetric.
 <input>u = vector(QQbar, [2+3*I,  5+2*I, -3+I])
-Again, notice that the result is the conjugate of what we have in the text.<br /><br />
 Norms are as easy as conjugates.  Easier maybe.  It might be useful to realize that Sage uses entirely distinct code to compute an exact norm over <code>QQbar</code> versus an approximate norm over <code>CDF</code>, though that is totally transparent as you issue commands.  Here is <acroref type="example" acro="CNSV" /> reprised.