This is pdfTeX, Version 3.1415926-1.40.9 (MiKTeX 2.7) (preloaded format=latex 2008.12.3) 2 DEC 2009 15:37 entering extended mode **final_rev_sol.tex (final_rev_sol.tex LaTeX2e <2005/12/01> Babel and hyphenation patterns for english, dumylang, nohyphenation, ge rman, ngerman, german-x-2008-06-18, ngerman-x-2008-06-18, french, loaded. ("C:\Program Files\MiKTeX 2.7\tex\latex\base\article.cls" Document Class: article 2005/09/16 v1.4f Standard LaTeX document class ("C:\Program Files\MiKTeX 2.7\tex\latex\base\size12.clo" File: size12.clo 2005/09/16 v1.4f Standard LaTeX file (size option) ) \c@part=\count79 \c@section=\count80 \c@subsection=\count81 \c@subsubsection=\count82 \c@paragraph=\count83 \c@subparagraph=\count84 \c@figure=\count85 \c@table=\count86 \abovecaptionskip=\skip41 \belowcaptionskip=\skip42 \bibindent=\dimen102 ) ("C:\Program Files\MiKTeX 2.7\tex\latex\amsfonts\amssymb.sty" Package: amssymb 2002/01/22 v2.2d ("C:\Program Files\MiKTeX 2.7\tex\latex\amsfonts\amsfonts.sty" Package: amsfonts 2001/10/25 v2.2f \@emptytoks=\toks14 \symAMSa=\mathgroup4 \symAMSb=\mathgroup5 LaTeX Font Info: Overwriting math alphabet `\mathfrak' in version `bold' (Font) U/euf/m/n --> U/euf/b/n on input line 132. )) ("C:\Program Files\MiKTeX 2.7\tex\latex\base\latexsym.sty" Package: latexsym 1998/08/17 v2.2e Standard LaTeX package (lasy symbols) \symlasy=\mathgroup6 LaTeX Font Info: Overwriting symbol font `lasy' in version `bold' (Font) U/lasy/m/n --> U/lasy/b/n on input line 47. ) ("C:\Program Files\MiKTeX 2.7\tex\latex\amsmath\amsmath.sty" Package: amsmath 2000/07/18 v2.13 AMS math features \@mathmargin=\skip43 For additional information on amsmath, use the `?' option. 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LaTeX Font Info: Redeclaring font encoding OMS on input line 568. \macc@depth=\count92 \c@MaxMatrixCols=\count93 \dotsspace@=\muskip10 \c@parentequation=\count94 \dspbrk@lvl=\count95 \tag@help=\toks16 \row@=\count96 \column@=\count97 \maxfields@=\count98 \andhelp@=\toks17 \eqnshift@=\dimen106 \alignsep@=\dimen107 \tagshift@=\dimen108 \tagwidth@=\dimen109 \totwidth@=\dimen110 \lineht@=\dimen111 \@envbody=\toks18 \multlinegap=\skip44 \multlinetaggap=\skip45 \mathdisplay@stack=\toks19 LaTeX Info: Redefining \[ on input line 2666. LaTeX Info: Redefining \] on input line 2667. ) ("C:\Program Files\MiKTeX 2.7\tex\latex\amsmath\amscd.sty" Package: amscd 1999/11/29 v1.2d \athelp@=\toks20 \minaw@=\dimen112 \bigaw@=\dimen113 \minCDarrowwidth=\dimen114 ) ("C:\Program Files\MiKTeX 2.7\tex\latex\tools\theorem.sty" Package: theorem 1995/11/23 v2.2c Theorem extension package (FMi) \theorem@style=\toks21 \theorem@bodyfont=\toks22 \theorempreskipamount=\skip46 \theorempostskipamount=\skip47 ("C:\Program Files\MiKTeX 2.7\tex\latex\tools\thp.sty" File: thp.sty 1995/11/23 v2.2c Theorem extension package (FMi) )) (final_rev_sol.aux) LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 38. LaTeX Font Info: ... okay on input line 38. LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 38. LaTeX Font Info: ... okay on input line 38. LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 38. LaTeX Font Info: ... okay on input line 38. LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 38. LaTeX Font Info: ... okay on input line 38. LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 38. LaTeX Font Info: ... okay on input line 38. LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 38. LaTeX Font Info: ... okay on input line 38. LaTeX Font Info: Try loading font information for U+msa on input line 49. ("C:\Program Files\MiKTeX 2.7\tex\latex\amsfonts\umsa.fd" File: umsa.fd 2002/01/19 v2.2g AMS font definitions ) LaTeX Font Info: Try loading font information for U+msb on input line 49. ("C:\Program Files\MiKTeX 2.7\tex\latex\amsfonts\umsb.fd" File: umsb.fd 2002/01/19 v2.2g AMS font definitions ) LaTeX Font Info: Try loading font information for U+lasy on input line 49. ("C:\Program Files\MiKTeX 2.7\tex\latex\base\ulasy.fd" File: ulasy.fd 1998/08/17 v2.2e LaTeX symbol font definitions ) Overfull \hbox (0.33568pt too wide) in paragraph at lines 76--93 \OML/cmm/m/it/12 x \OT1/cmr/m/n/12 = [] = \OML/cmm/m/it/12 x[] [] \OT1/cmr/m/n/ 12 + \OML/cmm/m/it/12 x[] []$\OT1/cmr/m/n/12 . Thus $\OMS/cmsy/m/n/12 f\OML/cmm /m/it/12 w[] \OT1/cmr/m/n/12 = [] \OML/cmm/m/it/12 ; w[] \OT1/cmr/m/n/12 = [] Overfull \hbox (8.51372pt too wide) in paragraph at lines 120--125 [][]\OT1/cmr/m/n/12 Hence $\OMS/cmsy/m/n/12 f\OML/cmm/m/it/12 v[] \OT1/cmr/m/n/ 12 = [] \OML/cmm/m/it/12 ; v[] \OT1/cmr/m/n/12 = []\OMS/cmsy/m/n/12 g$\OT1/cmr/ m/n/12 is an or-thog-o-nal ba-sis for $\OML/cmm/m/it/12 Null\OT1/cmr/m/n/12 (\O ML/cmm/m/it/12 A \OMS/cmsy/m/n/12 [] [1 ] Overfull \hbox (18.548pt too wide) in paragraph at lines 127--149 [] [] [] []$ $\OML/cmm/m/it/12 x \OMS/cmsy/m/n/12 2 \OML/cmm/m/it/12 Null\OT1/c mr/m/n/12 (\OML/cmm/m/it/12 A \OMS/cmsy/m/n/12 [] [2] Overfull \hbox (0.4336pt too wide) in paragraph at lines 173--181 [][]\OT1/cmr/m/n/12 Finally, we have $\OML/cmm/m/it/12 A \OT1/cmr/m/n/12 = \OML /cmm/m/it/12 P [] P[]$ \OT1/cmr/m/n/12 where $\OML/cmm/m/it/12 P \OT1/cmr/m/n/1 2 = [[] [] []] = [] Overfull \hbox (4.03725pt too wide) in paragraph at lines 184--186 [][]\OT1/cmr/m/n/12 Solution: Re-call that the quadratic form in $\OML/cmm/m/it /12 x[]; x[]$ \OT1/cmr/m/n/12 and $\OML/cmm/m/it/12 x[]$ \OT1/cmr/m/n/12 is $\O ML/cmm/m/it/12 Q[]\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 x\OT1/cmr/m/n/12 ) = [] Overfull \hbox (18.40099pt too wide) in paragraph at lines 237--248 [][]\OT1/cmr/m/n/12 What is $\OML/cmm/m/it/12 lim[]A[]$\OT1/cmr/m/n/12 ? Re-cal l that $\OML/cmm/m/it/12 A \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 PDP[]$ \OT1/cmr/m /n/12 and $\OML/cmm/m/it/12 A[] \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 PD[]P[] \OT1 /cmr/m/n/12 = [] Overfull \hbox (26.31635pt too wide) in paragraph at lines 237--248 \OML/cmm/m/it/12 P [] P[] \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 P [] P[]$\OT1/cmr/ m/n/12 . Note that $[][] 4[] = [] [3] Overfull \hbox (6.33055pt too wide) in paragraph at lines 297--306 \OT1/cmr/m/n/12 an or-thonor-mal ba-sis of eigen-vec-tors. So we have $\OML/cmm /m/it/12 A \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 P [] P[]$ [] [4] Overfull \hbox (29.69157pt too wide) in paragraph at lines 330--339 [][]$\OML/cmm/m/it/12  \OT1/cmr/m/n/12 = \OMS/cmsy/m/n/12 \OT1/cmr/m/n/12 1$: $\OML/cmm/m/it/12 A \OMS/cmsy/m/n/12 \OT1/cmr/m/n/12 (\OMS/cmsy/m/n/12 \OT1 /cmr/m/n/12 1) \OMS/cmsy/m/n/12  \OML/cmm/m/it/12 I \OT1/cmr/m/n/12 = [] = [] [] []$. [] Overfull \hbox (10.83029pt too wide) in paragraph at lines 340--349 [][]$\OML/cmm/m/it/12  \OT1/cmr/m/n/12 = \OMS/cmsy/m/n/12 \OT1/cmr/m/n/12 6$: $\OML/cmm/m/it/12 A \OMS/cmsy/m/n/12 \OT1/cmr/m/n/12 (\OMS/cmsy/m/n/12 \OT1 /cmr/m/n/12 6) \OMS/cmsy/m/n/12  \OML/cmm/m/it/12 I \OT1/cmr/m/n/12 = [] = [] [] []$. [] Overfull \hbox (19.12233pt too wide) in paragraph at lines 350--361 \OT1/cmr/m/n/12 an or-thonor-mal ba-sis of eigen-vec-tors. So we have $\OML/cmm /m/it/12 A \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 P [] P[]$ [] [5] [6] Underfull \hbox (badness 10000) in paragraph at lines 436--437 [] [7] [8] Overfull \hbox (1.78166pt too wide) in paragraph at lines 490--497 \OML/cmm/m/it/12 A$ \OT1/cmr/m/n/12 are in-de-pen-dent. So $\OML/cmm/m/it/12 Co l\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 A\OT1/cmr/m/n/12 ) = \OML/cmm/m/it/12 span\O MS/cmsy/m/n/12 f[] \OML/cmm/m/it/12 ; [] ; []\OMS/cmsy/m/n/12 g$\OT1/cmr/m/n/12 . [] Overfull \hbox (1.59134pt too wide) in paragraph at lines 515--526 [] []$ $[] []$ [] Overfull \hbox (66.29224pt too wide) in paragraph at lines 515--526 [] [] [] []$ [] [9] [10] Overfull \hbox (52.39912pt too wide) in paragraph at lines 620--630 \OMS/cmsy/m/n/12 jj\OML/cmm/m/it/12 v[]\OMS/cmsy/m/n/12 jj \OT1/cmr/m/n/12 = [] = []$. Hence $\OMS/cmsy/m/n/12 f[]\OML/cmm/m/it/12 ; []; []\OMS/cmsy/m/n/12 g \OT1/cmr/m/n/12 = \OMS/cmsy/m/n/12 f[] \OML/cmm/m/it/12 ; [] ; []\OMS/cmsy/m/n/ 12 g$ [] [11] Overfull \hbox (56.9479pt too wide) in paragraph at lines 714--732 \OT1/cmr/m/n/12 [\OML/cmm/m/it/12 A []\OT1/cmr/m/n/12 ] = [] [] []$ [] [12] [13] [14] Package amsfonts Warning: Obsolete command \Bbb; \mathbb should be used instead on input line 803. Underfull \hbox (badness 10000) in paragraph at lines 802--804 [] Underfull \hbox (badness 10000) in paragraph at lines 817--819 [] [15] Overfull \hbox (2.71826pt too wide) in paragraph at lines 838--850 [][]\OT1/cmr/m/n/12 Solution: Con-sider the aug-mented ma-trix $[\OML/cmm/m/it/ 12 A I\OT1/cmr/m/n/12 ] = []$ [] Overfull \hbox (64.65486pt too wide) in paragraph at lines 838--850 [] []$ $[] []$ [] [16] Overfull \hbox (28.4302pt too wide) in paragraph at lines 854--857 \OT1/cmr/m/n/12 Solution: The co-or-di-nate is $\OML/cmm/m/it/12 x \OT1/cmr/m/n /12 = \OML/cmm/m/it/12 A[] [] \OT1/cmr/m/n/12 = [] [] = [] Underfull \hbox (badness 10000) in paragraph at lines 902--918 [] [17] [18] Overfull \hbox (19.10275pt too wide) in paragraph at lines 987--1006 \OMS/cmsy/m/n/12 $ \OT1/cmr/m/n/12 ($\OML/cmm/m/it/12 r[] \OT1/cmr/m/n/12 := \OML/cmm/m/it/12 r[] \OMS/cmsy/m/n/12 \OML/cmm/m/it/12 r[]$\OT1/cmr/m/n/12 ) $[]$ $\OMS/cmsy/m/n/12 $ \OT1/cmr/m/n/12 ($\OML/cmm/m/it/12 r[] \OT1/cmr/m/n /12 := \OML/cmm/m/it/12 r[] \OMS/cmsy/m/n/12 \OML/cmm/m/it/12 r[]$\OT1/cmr/m/ n/12 ) $[]$ [] Overfull \hbox (41.97748pt too wide) in paragraph at lines 987--1006 \OMS/cmsy/m/n/12 $ \OT1/cmr/m/n/12 ($\OML/cmm/m/it/12 r[] \OT1/cmr/m/n/12 := \OML/cmm/m/it/12 r[] \OMS/cmsy/m/n/12 \OML/cmm/m/it/12 r[]$\OT1/cmr/m/n/12 ) $[]$$\OMS/cmsy/m/n/12 $ \OT1/cmr/m/n/12 ($\OML/cmm/m/it/12 r[] \OT1/cmr/m/n/ 12 := \OML/cmm/m/it/12 r[] \OT1/cmr/m/n/12 + 3\OML/cmm/m/it/12 r[]$\OT1/cmr/m/n /12 ) $[]$. [] [19] Overfull \hbox (86.49062pt too wide) in paragraph at lines 1008--1015 [][]\OT1/cmr/m/n/12 Let $\OML/cmm/m/it/12 A \OT1/cmr/m/n/12 = []$ which is row re-duced to $[]$ [] Overfull \hbox (25.38194pt too wide) in paragraph at lines 1029--1048 [][]\OT1/cmr/m/n/12 Solution: Con-sider the aug-mented ma-trix $[]$ [] [20] Overfull \hbox (96.84772pt too wide) in paragraph at lines 1049--1062 [][]\OT1/cmr/m/n/12 So the first, third and fifth vec-tor forms a ba-sis for Co l(A), i.e $\OMS/cmsy/m/n/12 f[] \OML/cmm/m/it/12 ; [] ; []\OMS/cmsy/m/n/12 g$ [] [21] Package amsmath Warning: Cannot use `split' here; (amsmath) trying to recover with `aligned' on input line 1124. Overfull \hbox (100.41945pt too wide) in alignment at lines 1106--1125 [][][] [] [] [22] Underfull \hbox (badness 10000) in paragraph at lines 1142--1147 [] [23] (final_rev_sol.aux) ) Here is how much of TeX's memory you used: 1205 strings out of 95305 12877 string characters out of 1183059 76758 words of memory out of 1500000 4426 multiletter control sequences out of 110000 9967 words of font info for 39 fonts, out of 3000000 for 5000 14 hyphenation exceptions out of 8191 27i,18n,24p,446b,279s stack positions out of 5000i,500n,10000p,200000b,50000s Output written on final_rev_sol.dvi (23 pages, 98996 bytes).