/Ascent 686 endobj >> << LupJ6T[SgKP4kfq+u0FJMAn&\goGa"G7""V"H*-6PLg8V%SH6d_FQu:f[8Q&(bfkH@[VD@^ElY@MA>j8X1miP6QZUe 0000046172 00000 n /StemV 76 /FirstChar 12 /XHeight 697 >> =G8&]>t`%B!(Pji1F8Oa.:MH6?5A+QQ$F)kh,J.Q>rQ6a,q\<2[MD1V?C;7+,a*<7U^R2f(\nLQ:gVLZCFmN! /Name/F3 rKRhpM!oP2*bZ2E16_k4N\?]0kW+/*71,'5GrF0+`13UBo!ubl3? :,] >> Orthogonal Projections. #8R9[1n$i_^Zigtp6gR4oTk@b(Q%tJeiY1[/VB,d&b[\"bsTbe2eG2=oh3&>*lXIS? =G8&]D+IT6! /Length1 738 << 44 0 obj N62(k9bIB@ "EMM4lRC$ZIf%JIgY0`U]nFZG-RSjo_kWK8pVq_>jdP;/.`rBX* =G=.c>Ar7c(4Q"]cm,s&R(3_T8II@LMtI.`kjB?k"Eki"M7-RNP#?DNoZuuTB.ali3>.4\1@,:@?&AhUtn;i /LastChar 121 /Subtype/Type1 ?R)@CN77HJe8/^9:6I@D1aXPKPqf/%jOG^$%IpNZ&i$&*0rNf9>m /Encoding 45 0 R e^bm'lWR&+G#k0M0EKpa*Q1BYUo-=0G /Filter[/ASCII85Decode/FlateDecode] Efk,Lg^#PtY4J5,("-^)bgkh=AA.urqK'=:c0us//GAi4`+-(7k8>rYT'LW?pdc/Mc7p*tlM8M[ e^1`#ancG>*6uZ][W[NeBBUNkDbjHJ3i,f6+HX12Y+/@PdVXIEUXC':D,7F9Z7#a+!`mH8+V:VejF(gga"$B>. >> (0os6$N,T^l-lDEsfjn[>QL&%,iAM.5t]Wt-]/)=PVc^;3>KoF6p endobj /Type/Font 58 0 obj /CapHeight 683 /LastChar 121 /Subtype/Type1C /MediaBox[0 0 612 792] >> /Name/F10 17 0 obj !1#KR]jfWE0gEDII%/6qbHlB3YM*2*hSe /Name/F0 /Subtype/Type1C /CapHeight 683 endobj /Type/Encoding /Type/FontDescriptor /Flags 34 /FontDescriptor 11 0 R 0000052747 00000 n /Descent -194 YkVc\Sg8j`4+*fU\]*')eTdgrojAdA-Y157-Cpn=ZL(.*d7uY7,WG3jJCfmO)'5$16WumqHC+`. a/>jeXUq,igM%+ofs69"]oY6hVG+)CXQg1;Mn;^cZ_ mathsfreak. 7iTq[)@a@ap`3o93F#TqLho>e&N:<4''D`;82P1eIi"QQWG)RQ_`%%&Y#oQ'#PI$6Rn^=jY2io^ '.O\I5qp5l,)2JdLa#k20+uHaKu?mK3T:W[%IjM)MKk@f:+RN"&.-;H6mD.40p]>AGa=.5jq2UO 30[I*UX^+?0Nu0(nL1`mI*V0j%>2Fk)6=m]\3A*OMQnNaEMJMF3c;0`PB(S?Gm-a*.4n.\Yme>l /Encoding 57 0 R /Descent -194 /Type/Font 5':Y0;.6o?GBB;a)KN4/dJaHFUX,K(#[jLi:=U-1`q4D:Jt4"f4EWUS3g^$p-Tb+;-"b7QBETF/ >> 30 0 obj "`2)Lad#kFAkuheb7+=@er#1=YCC9P^sYVUjF.X"b5r 9 0 obj /Widths[568 568 568 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 637 0 0 0 0 0 0 0 0 0 0 0 447] 40 0 obj ^:EOMBF9,CDF?h8Q/;>6t6p]Hg.RZlPZ3qc_:4$f3\=o_I:CU>_4nmbZP 6Y_`iL"EI.GqQDYg]+,B.P8lic"Ft>=FN+$Slnn+,mq3Wi>.\[R*6#]SX,>N`:.N$TZBnS6k2t-Jm_aUWtj38)LWWEQ2Yf6Xd9%9bBI+)XC J^7TI8p?Kr%i4:K!2l9Q^b_3^5\>CgXF9H[9@r)?m=UPaD#9j%?5FWNWaMjcQ\(jmo`$bV_kQ29 /FontDescriptor 51 0 R /Type/Encoding /Subtype/Type1C /StemV 85 endobj /XHeight 704 endobj For the second claim, note that if A~z=~0, then stream /ItalicAngle -14 aY;YVA?K\#mM[i@'A7EX!.\Tu=EhBW[grU0RC0Yh'kK"n+_6];:)8TW ?4AfX,;/.Y+#h'"hBSg$2V242f#:b!E`l$7?3IC%Gb-9r!`JX06RX)(\s\1$,VO^H.m/9eh29sb@FCJF startxref /ItalicAngle 0 /StemV 114 /Widths[578 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 277 0 277 0 0 0 0 0 0 0 0 0 0 0 0 0 0 783 0 0 0 0 0 0 0 687 0 0 0 361 0 0 0 0 0 0 687 783 735 0 723 747 0 0 0 0 0 0 0 0 0 0 0 506 0 446 554 446 301 506 554 277 0 0 277 831 554 506 554 0 397 397 385 554 530 723 530 530] /CapHeight 655 A projection on a vector space is a linear operator : → such that =.. Example: Prove Q = \(\begin{bmatrix} cosZ & sinZ \\ -sinZ & cosZ\\ \end{bmatrix}\) is orthogonal matrix. XbbQ^Zg#ZppXEjn)&c6`+W.jP@t5%XO>a81M%W3+cK6&P!_$[NUg^ /StemV 78 Click here to toggle editing of individual sections of the page (if possible). ^t79*16u5.7C#1ipGE%/%u1bn0JCkSZ*AkBF]a=[,#p;P:r$GRWdtK7?%')srWhd71)An>CFO&Q /Type/FontDescriptor /Type/Encoding Since det (A) = det (Aᵀ) and the determinant of product is the product of determinants when A is an orthogonal matrix. @Nt'L^ij?8pZ,II*H]O3A/\->Is%W"q\\6?OgF[8j'bsNi@)&Mq65eSk,?j#%g>o stream stream 63%1a-ToGenYjQ3J4F&5QROU*L-PYT,Rb1'Y*>hSAS2i!YofNSQFrjP!9T6K>ad;:k?X/;AT#-)N/7TM#iH(0;L[FIJlc7/]W[-ll41r'8L*.4S==am8<61q"8\)8Cnr4a73J0!t)J2mR`8U@=X_?u2bmP"A=S%!/6] RJXPO9J3+fZHQA]P"\RsXGhP'FG`fLnC\>bAbijmXK(NJO(p+WVI3>8Q$#B!CDN1l\&Pm$pF;!F 2 0 obj endobj )0iqMm8d5 ,s;19jmK+9H&NgGJ7)`!Vcs5_b [Y2pRm8pB0& /BaseFont/KXBIAH+cmr5 /Parent 3 0 R << ;ESk&?AqB`>:(d:WDui-Ak$tCh(M%a3HH%=$:/o8.fSheRq!XKdGQ#gaLETs b^0Ek:l7m+e0d"@a@&@t#=*,eOm 0000052965 00000 n /Subtype/Type1 )t4Ulib)bP/AM;8!LQ;oUKtsS'6PL@ /BaseEncoding/MacRomanEncoding >> /ID[<0E809E80EA7D6CDEEE455D5A13DB7968><0E809E80EA7D6CDEEE455D5A13DB7968>] pslt0"A6]B`R"a75069*:q/o);"'#[nN-FXqS>+Q^9m=Y\tK>)@S"VCVE9'HU([FoYd;.1)S3,o-0Zp@MC@#?J=o,ol@a,gsEV^=0]k_"1.1de&"s&2o,Ig /Subtype/Type1 d>ZL)GDKH#m06gEr_%aI95WXX$7h9e6D*3LBia:mohN7shJO6@f-"*oF,EXHpMn3f])4gJPrVVC /Encoding 25 0 R 0000053462 00000 n /Filter[/ASCII85Decode/FlateDecode] View wiki source for this page without editing. 19 0 obj stream Theorem Let A be an m × n matrix, let W = Col ( A ) , and let x be a vector in R m . ;>O="\:1A(iblj-II=PI/B4oZt88lFk2CcplAJnYo*RW3>?g[)u9(#emGD 0WN2lVcN>rbsq@KGbMR# )l/n#Wm&8/"TLs)Zh3b7cFkh,p$Ck9>;-=CDHshc&hBQ*?-G"QaOtc*Qu&CeLXI,bP&Fb@gnbM4 >> /ItalicAngle 0 hmfcK1dc.XJFAB_gO7\6=60.Z2<5TJq1b+m9h&Z)m)[@-R!_tHV&2mokVWg&5"*5#1Q"Qc;#lJA &0jgFj<7m>cA^%EOsMkZ]SGK1q+Z%5Lud<4]`Jk$h2@\#2SZP$atQ>rc1cRFd^h7UHT%WoZ>"H)-oDgUQ$bW:5;g%u $O)`Fh1Z2VH/(i$UV,kZk.UXD>^+B1nhtY&4[5!/57Be`fHiD"?Z.c61Y:0cBeDfeQ+D$BiJJEU /BaseEncoding/MacRomanEncoding Ut"_'4TH'H:_u4+=`,kK? << Find out what you can do. endobj /FirstChar 110 0000000000 65535 f endobj /Type/Encoding 0000043343 00000 n endobj _fh*k8MOH0d^!M2/DRtq,Kr=reKZUt2(g1"0(uQ6Min]rUo22.G_@?]`has'>E\&'"Ru[.5_K]? H4;6"[DPdt#+lSJDEUjfp#eoq_MV+nSlH9E6D%+,YB3e@(i=+[&1f'?fn^X 53529 /Ascent 683 /Length 4687 )7*[l]k,kB4D&;`LZeT@ci)OoFL,tjZoQ>A&fI)#1&ZkB+9:rFHV^fJ])#GR,AuJHZF:%`p+mU, << /Widths[816 0 0 0 0 0 0 0 0 0 0 1094 0 0 0 0 0 641 802 0 0 0 0 0 0 0 0 0 0 0 0 558 0 509 0 530 0 0 641 321 0 0 0 955 0 572 0 0 474 453 446 0 0 0 606] \begin{align} \vec{u} \cdot \vec{b} = (k\vec{b} + \vec{w_2}) \cdot \vec{b} \\ \vec{u} \cdot \vec{b} = k(\vec{b} \cdot \vec{b}) + \vec{w_2} \cdot \vec{b} \\ \vec{u} \cdot \vec{b} = k \| \vec{b} \|^2 \\ k = \frac{\vec{u} \cdot \vec{b}}{\| \vec{b} \|^2} \end{align}, \begin{align} \vec{w_1} =\mathrm{proj}_{\vec{b}} \vec{u} = \frac{(\vec{u} \cdot \vec{b})}{\| \vec{b} \|^2} \vec{b} \\ \blacksquare \end{align}, \begin{align} \| \mathrm{proj}_{\vec{b}} \vec{u} \| = \biggr \| \frac{(\vec{u} \cdot \vec{b})}{\| \vec{b} \|^2} \vec{b} \biggr \| \\ \| \mathrm{proj}_{\vec{b}} \vec{u} \| = \mathrm{abs}\left ( \frac{(\vec{u} \cdot \vec{b})}{\| \vec{b} \|^2} \right ) \| \vec{b} \| \\ \| \mathrm{proj}_{\vec{b}} \vec{u} \| = \frac{\mid \vec{u} \cdot \vec{b}\mid}{\| \vec{b} \|^2} \| \vec{b} \| \\ \| \mathrm{proj}_{\vec{b}} \vec{u} \| = \frac{\mid \vec{u} \cdot \vec{b}\mid}{\| \vec{b} \|} \\ \| \mathrm{proj}_{\vec{b}} \vec{u} \| = \frac{\mid \| \vec{u} \| \| \vec{b} \| \cos \theta \mid}{\| \vec{b} \|} \\ \| \mathrm{proj}_{\vec{b}} \vec{u} \| = \frac{\| \vec{u} \| \| \vec{b} \| \mid \cos \theta \mid}{\| \vec{b} \|} \\ \| \mathrm{proj}_{\vec{b}} \vec{u} \| = \mid \cos \theta \mid \| \vec{u} \| \quad \blacksquare \end{align}, Unless otherwise stated, the content of this page is licensed under. 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Proof. /Type/Font s,+h1M9%`g?VP.hDR;.I?CFfYWEJt#4!H^J4jFtseO6*dH`kp&24%`LI]d(`BoO-[CBcNHYq_>W&\&`.>TnK"b_;'#a*:BtBlGJX^WfJ@1ajPI? /Type/Encoding /Length 5252 xref +r)6Pdeme>8Y^P4+nA$@4`JT0pNp:>&)MXSQ@I8*Sq-.1;pnippG%f0^-UAda9MsM9_=mtlgGoK /Length1 361 [bY\G!pCWPVX&AR#ePWDoc\9;XQX_+Mrg(IEk]NEjE6]F5#sN__GWc=\H@]ioqtE];Yhu!7ZkU( endobj 29 0 obj 0000044931 00000 n 38 0 obj H2>aPf&jk+$Y*W9X;9G'>T]uVRI%&fJ9c>a0/J_+jFVkJng@+'&pXVT.! View/set parent page (used for creating breadcrumbs and structured layout). !m#Rh*Gq'`-4?e#bc\I)]?.q3e2JZ jlg"g1XLtWU1'85_SIZa6;@mXPYmJk+)ZIV7q%n]`q*!jZcRS$M%>-@,!T(oSt]8_WnHqA%VQZl /Differences[66/B 77/M 83/S/T 97/a 99/c 101/e 104/h/i 109/m 111/o 114/r/s/t 120/x] :j%ApFE_'W. Kr`Tb$hoqr2h"*_lM? 18\!0,]Z7ObIWdmDXC7@._)g>;_m3i@sdGFkNo9. @ 1C5B>Y%192'OUrTi2qiL8[EX_8'#Y.`$T^r^_%'? << /BaseEncoding/MacRomanEncoding (Pji?8l.kqinn(Hf(Ga+RYT!59]c&f#h#RPC[#J(S0p\g:c+sQljO6S)]d7DHsXT /Ascent 683 /Parent 3 0 R /LastChar 120 /StemV 93 d:aO-7l+!V>)h+++>eA6(k_D+qW[5>-p8e3hQXDo*S2mo7U@]$ ooWk?oHVP4! 2#@uhhJXHYqe]PU'#bEJE@`T"D035E,lHLY+7_3/&Q5bl@B'XN? >> ProofSince is orthogonal, the columns are in, soT orthonormal vectors ‘# +-œ"œ,. ?f\U!+UnL$AdomnPo^8>H/hB?QYk=X:'1k:c++tUb;#DYB$!l"Q1<= 0%U9m"ptK0fcj+Xfh+\l]-nep#iA1CXKh1c4Mt;5RBH1@KfA?sqp%Ji4J. 6'tmeK6!C/4ERckGJND&!nfsa)m[a/Sp.s4#atoN#lg$ZSdPOCI4WA)j!h],0qTGUeWYN=+DK9p Q1XcsfC!)o7McTh^Y&Z->)]hiE=_UsdVWej.-/$hoOaJt(m#-!H7@>\dEEVd. :cKAQTGWI',+C*X6r#@B/gl]S$"\@iLuh57`C_G@[MoiO@gScEm6!36qo/1)endstream The vector $ \vec { u } $ be a linear operator: such...,  category ) of the page ( used for creating breadcrumbs and structured layout ) GrY/-7'^4 \XOS6. 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