This is pdfTeX, Version 3.1415926-1.40.9 (MiKTeX 2.7) (preloaded format=pdflatex 2008.12.3) 3 NOV 2009 12:07 entering extended mode **review2_sol.tex (review2_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) )) (review2_sol.aux) LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 11. LaTeX Font Info: ... okay on input line 11. LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 11. LaTeX Font Info: ... okay on input line 11. LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 11. LaTeX Font Info: ... okay on input line 11. LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 11. LaTeX Font Info: ... okay on input line 11. LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 11. LaTeX Font Info: ... okay on input line 11. LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 11. LaTeX Font Info: ... okay on input line 11. LaTeX Font Info: Try loading font information for U+msa on input line 23. ("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 23. ("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 23. ("C:\Program Files\MiKTeX 2.7\tex\latex\base\ulasy.fd" File: ulasy.fd 1998/08/17 v2.2e LaTeX symbol font definitions ) Underfull \hbox (badness 10000) in paragraph at lines 23--24 [] [1 {C:/Documents and Settings/All Users/Application Data/MiKTeX/2.7/pdftex/config/ pdftex.map}] Overfull \hbox (3.84732pt too wide) in paragraph at lines 83--86 []\OT1/cmr/m/n/12 Solution: (a) Since $\OML/cmm/m/it/12 det\OT1/cmr/m/n/12 (\OM L/cmm/m/it/12 A\OT1/cmr/m/n/12 ) = \OMS/cmsy/m/n/12 \OT1/cmr/m/n/12 1$, we hav e $\OML/cmm/m/it/12 A[] \OT1/cmr/m/n/12 = [] [] = []$ [] Package amsmath Warning: Cannot use `split' here; (amsmath) trying to recover with `aligned' on input line 107. Overfull \hbox (22.95366pt too wide) in alignment at lines 87--108 [][][] [] [] Package amsmath Warning: Cannot use `split' here; (amsmath) trying to recover with `aligned' on input line 115. [2] Package amsmath Warning: Cannot use `split' here; (amsmath) trying to recover with `aligned' on input line 138. [3] Overfull \hbox (68.93134pt too wide) in paragraph at lines 159--181 []$ $[]$ $[]$ $[]$\OT1/cmr/m/n/12 . [] Underfull \hbox (badness 10000) in paragraph at lines 159--181 [] Overfull \hbox (25.90242pt too wide) in paragraph at lines 206--216 \OT1/cmr/m/n/12 Let $\OML/cmm/m/it/12 A \OT1/cmr/m/n/12 = []$. So $\OML/cmm/m/i t/12 x \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 A\OT1/cmr/m/n/12 [\OML/cmm/m/it/12 x\ OT1/cmr/m/n/12 ][] = [] [] = [] = [] [4] Overfull \hbox (26.77705pt too wide) in paragraph at lines 223--243 [] []$ $[] []$ $[] [] [] [5] Overfull \hbox (75.20161pt too wide) in paragraph at lines 294--310 []\OT1/cmr/m/n/12 Find a ba-sis for the sub-space spanned by the fol-low-ing ve c-tors $\OMS/cmsy/m/n/12 f[] \OML/cmm/m/it/12 ; [] ; [] ; []\OMS/cmsy/m/n/12 g$ \OT1/cmr/m/n/12 . [] [6] Underfull \hbox (badness 10000) in paragraph at lines 357--368 [] [7] Overfull \hbox (64.76505pt too wide) in paragraph at lines 419--436 \OML/cmm/m/it/12 A \OMS/cmsy/m/n/12 \OML/cmm/m/it/12 I \OT1/cmr/m/n/12 = []$ $= []$ $[] []$. [] Underfull \hbox (badness 10000) in paragraph at lines 419--436 [] Overfull \hbox (24.2673pt too wide) in paragraph at lines 437--466 \OT1/cmr/m/n/12 The ba-sis for the eigenspace cor-re-spond-ing to eigen-value $ 1$ is $\OMS/cmsy/m/n/12 f[] \OML/cmm/m/it/12 ; []\OMS/cmsy/m/n/12 g$ [] Underfull \hbox (badness 10000) in paragraph at lines 501--502 [] [8] Underfull \hbox (badness 10000) in paragraph at lines 521--531 [] Underfull \hbox (badness 10000) in paragraph at lines 532--544 [] Underfull \hbox (badness 10000) in paragraph at lines 545--561 [] [9] Underfull \hbox (badness 10000) in paragraph at lines 565--582 [] Underfull \hbox (badness 10000) in paragraph at lines 602--619 [] Underfull \hbox (badness 10000) in paragraph at lines 652--653 [] [10] Underfull \hbox (badness 10000) in paragraph at lines 660--663 [] Overfull \hbox (12.99818pt too wide) in paragraph at lines 694--705 \OMS/cmsy/m/n/12  \OT1/cmr/m/n/12 (\OML/cmm/m/it/12 r[] \OT1/cmr/m/n/12 = \OMS /cmsy/m/n/12 \OML/cmm/m/it/12 r[]; r[] \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 r[] \OT1/cmr/m/n/12 + 2\OML/cmm/m/it/12 r[]\OT1/cmr/m/n/12 ) []$ $\OMS/cmsy/m/n/12  \OT1/cmr/m/n/12 (\OML/cmm/m/it/12 r[] \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 r[] \OT1/cmr/m/n/12 + 2\OML/cmm/m/it/12 r[]\OT1/cmr/m/n/12 ) []$ [] [11] Underfull \hbox (badness 10000) in paragraph at lines 721--724 [] Underfull \hbox (badness 10000) in paragraph at lines 729--731 [] [12] [13] (review2_sol.aux) ) Here is how much of TeX's memory you used: 1190 strings out of 95304 12780 string characters out of 1183032 72758 words of memory out of 1500000 4413 multiletter control sequences out of 110000 8919 words of font info for 36 fonts, out of 3000000 for 5000 14 hyphenation exceptions out of 8191 27i,18n,24p,225b,264s stack positions out of 5000i,500n,10000p,200000b,50000s Output written on review2_sol.pdf (13 pages, 86649 bytes). PDF statistics: 85 PDF objects out of 1000 (max. 8388607) 0 named destinations out of 1000 (max. 131072) 1 words of extra memory for PDF output out of 10000 (max. 10000000)