<div id="pf1" class="pf w0 h0" data-page-no="1"><div class="pc pc1 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://files.passeidireto.com/f576e19b-d334-4686-aa50-430d24758f0a/bg1.png"><div class="c x0 y1 w2 h2"><div class="t m0 x1 h3 y2 ff1 fs0 fc0 sc0 ls2f ws0">EA<span class="_0 blank"></span>E0<span class="_0 blank"></span>207<span class="_0 blank"></span>:<span class="_1 blank"> </span>M<span class="_0 blank"></span>ate<span class="_0 blank"></span>mát<span class="_0 blank"></span>ica A<span class="_0 blank"></span>pli<span class="_0 blank"></span>cad<span class="_0 blank"></span>a à Eco<span class="_0 blank"></span>no<span class="_0 blank"></span>mia</div><div class="t m0 x2 h4 y3 ff1 fs1 fc0 sc0 ls2f ws1">Aula 8:<span class="_2 blank"> </span>Conjuntos Gerado<span class="_0 blank"></span>res e Base e Dimensão</div><div class="t m0 x3 h4 y4 ff1 fs1 fc1 sc0 ls2f ws2">Ma<span class="_0 blank"></span>rcos Y. Nakaguma</div><div class="t m0 x4 h4 y5 ff1 fs1 fc1 sc0 ls2f ws3">25/08/2017</div><div class="t m0 x5 h5 y6 ff1 fs2 fc1 sc0 ls2f">1</div></div><div class="c x0 y7 w2 h2"><div class="t m0 x6 h3 y8 ff1 fs0 fc0 sc0 ls2f ws4">Re<span class="_0 blank"></span>vis<span class="_0 blank"></span>ão</div><div class="t m0 x7 h4 y9 ff1 fs1 fc1 sc0 ls2f ws5">Na aula passada, de\u2026nimos que um conjunto de veto<span class="_0 blank"></span>res <span class="ff2 ws3">v</span><span class="fs2 ls0 v1">1</span><span class="ff3 ls1">,</span><span class="ff2 ws3">v</span><span class="fs2 ls2 v1">2</span><span class="ff3 ws6">, ...,</span></div><div class="t m0 x7 h6 ya ff2 fs1 fc1 sc0 ls2f ws3">v<span class="ff4 fs2 ls3 v1">k</span><span class="ff5 fs3 ls4 v0">2<span class="ff6 fs1 ls5">R<span class="ff4 fs2 ls6 v2">n</span><span class="ff1 ls7">é<span class="fc2 ls2f ws7">linea<span class="_0 blank"></span>rmente<span class="_3 blank"> </span>indep endente<span class="_3 blank"> </span><span class="fc1 ws2">se, e somente se,</span></span></span></span></span></div><div class="t m0 x0 h7 yb ff4 fs1 fc1 sc0 ls8">c<span class="ff1 fs2 ls0 v1">1</span><span class="ff3 ls2f ws3">.<span class="ff2">v<span class="ff1 fs2 ls9 v1">1</span><span class="ff7 fs3 lsa">+</span></span></span><span class="lsb">c<span class="ff1 fs2 ls2 v1">2</span><span class="ff3 ls2f ws3">.<span class="ff2">v<span class="ff1 fs2 lsc v1">2</span><span class="ff7 fs3 lsa">+</span></span><span class="ws8">... <span class="ff7 fs3 lsa">+</span></span></span><span class="lsd">c<span class="fs2 lse v1">k</span><span class="ff3 ls2f ws3">.<span class="ff2">v</span></span><span class="fs2 lsf v1">k</span><span class="ff7 fs3 ls10">=</span><span class="ff2 ls2f">0</span></span></span></div><div class="t m0 x7 h8 yc ff1 fs1 fc1 sc0 ls2f ws9">somente quando <span class="ff4 lsd">c</span><span class="fs2 ls11 v1">1</span><span class="ff7 fs3 ls12 v0">=<span class="ff4 fs1 lsd">c</span></span><span class="fs2 ls13 v1">2</span><span class="ff7 fs3 ls12 v0">=<span class="ff4 fs1 lsd">c<span class="fs2 ls14 v1">k</span></span>=</span><span class="v0">0.</span></div><div class="t m0 x7 h9 yd ff1 fs1 fc1 sc0 ls2f ws5">Vimos que um conjunto de <span class="ff4 ls15">k</span><span class="wsa">veto<span class="_0 blank"></span>res <span class="ff2 ws3">v</span><span class="fs2 ls2 v1">1</span><span class="ff3 ls16">,</span><span class="ff2 ws3">v</span><span class="fs2 ls0 v1">2</span><span class="ff3 ws6">, ...,<span class="_3 blank"> </span><span class="ff2 ws3">v<span class="ff4 fs2 ls17 v1">k</span></span></span><span class="wsb">em <span class="ff6 ls18">R<span class="ff4 fs2 ls19 v2">n</span></span>é</span></span></div><div class="t m0 x7 h4 ye ff1 fs1 fc1 sc0 ls2f ws2">linea<span class="_0 blank"></span>rmente indep<span class="_4 blank"> </span>endente se, e somente se:</div><div class="t m0 x8 ha yf ff1 fs1 fc1 sc0 ls2f wsc">p osto <span class="ff7 fs3 ls1a v0">(</span><span class="ff4 ls1b v0">A<span class="ff7 fs3 ls1c v0">)<span class="ls1d v0">=</span></span><span class="ls1e">k</span></span><span class="wsd v0">(= n<span class="ff4 fs2 ls1f v2">o</span><span class="wse">vetores)<span class="_5 blank"> </span>(<span class="ff5 fs3 ls20">\ue003</span>)</span></span></div><div class="t m0 x7 hb y10 ff1 fs1 fc1 sc0 ls2f wsd">Além disso, dado <span class="ff4 ls21">n</span><span class="wsf">veto<span class="_0 blank"></span>res <span class="ff2 ws3">v</span><span class="fs2 ls0 v1">1</span><span class="ff3 ls1">,</span><span class="ff2 ws3">v</span><span class="fs2 ls22 v1">2</span><span class="ff3 ws6">, ...,<span class="_3 blank"> </span><span class="ff2 ws3">v<span class="ff4 fs2 ls23 v1">n</span></span></span><span class="wsb">em <span class="ff6 ls24">R<span class="ff4 fs2 ls25 v2">n</span></span><span class="ws10">,<span class="_3 blank"> </span>po demos<span class="_3 blank"> </span>dizer<span class="_6 blank"> </span>que</span></span></span></div><div class="t m0 x7 h4 y11 ff1 fs1 fc1 sc0 ls2f ws2">eles são linea<span class="_0 blank"></span>rmente indep<span class="_4 blank"> </span>endentes se, e somente se:</div><div class="t m0 x9 hc y12 ff1 fs1 fc1 sc0 ls2f ws11">det <span class="ff8 ls26 v3">\ue000</span><span class="ff2 ws3 v0">v</span><span class="fs2 ls27 v1">1</span><span class="ff2 ws3 v0">v</span><span class="fs2 ls28 v1">2</span><span class="ff5 fs3 ws12 v0">\ue001 \ue001 \ue001<span class="_7 blank"> </span><span class="ff2 fs1 ws3">v<span class="ff4 fs2 ls29 v1">n</span><span class="ff8 ls2a v3">\ue001</span></span><span class="ls2b v0">6<span class="ff7 ls2c">=</span></span></span><span class="wse v0">0<span class="_8 blank"> </span>(<span class="ff5 fs3 ws13">\ue003<span class="_4 blank"> </span>\ue003</span>)</span></div><div class="t m0 x7 hd y13 ff1 fs1 fc2 sc0 ls2f ws14">T<span class="_9 blank"></span>eo<span class="_0 blank"></span>rema: <span class="fc1 ws15">Se <span class="ff4 ls2d">k<span class="ff9 fs3 ls12">></span><span class="ls2e">n</span></span><span class="wsd">, qualquer conjunto de <span class="ff4 ls1e">k</span><span class="ws16">vetores em <span class="ff6 ls5">R<span class="ff4 fs2 ls6 v2">n</span></span>é</span></span></span></div><div class="t m0 x7 h4 y14 ff1 fs1 fc1 sc0 ls2f ws17">linea<span class="_0 blank"></span>rmente<span class="_3 blank"> </span>dep endente</div><div class="t m0 x5 h5 y15 ff1 fs2 fc1 sc0 ls2f">2</div></div><div class="c x0 y16 w2 h2"><div class="t m0 xa he y17 ff1 fs4 fc1 sc0 ls2f ws18">C<span class="_9 blank"></span>o<span class="_0 blank"></span>n<span class="_0 blank"></span>ju<span class="_9 blank"></span>n<span class="_0 blank"></span>t<span class="_0 blank"></span>o<span class="_0 blank"></span>s G<span class="_0 blank"></span>e<span class="_0 blank"></span>r<span class="_0 blank"></span>a<span class="_0 blank"></span>d<span class="_0 blank"></span>o<span class="_9 blank"></span>r<span class="_0 blank"></span>e<span class="_0 blank"></span>s</div><div class="t m0 x5 h5 y18 ff1 fs2 fc1 sc0 ls2f">3</div></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf2" class="pf w0 h0" data-page-no="2"><div class="pc pc2 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://files.passeidireto.com/f576e19b-d334-4686-aa50-430d24758f0a/bg2.png"><div class="c x0 y1 w2 h2"><div class="t m0 x6 h3 y19 ff1 fs0 fc0 sc0 ls2f ws1a">Co<span class="_0 blank"></span>nju<span class="_0 blank"></span>nt<span class="_0 blank"></span>os Ge<span class="_0 blank"></span>rad<span class="_0 blank"></span>o<span class="_0 blank"></span>res</div><div class="t m0 x7 h9 y1a ff1 fs1 fc2 sc0 ls2f ws1b">De\u2026nição: <span class="fc1 ws1c">Seja <span class="ff2 ws3">v</span><span class="fs2 ls2 v1">1</span><span class="ff3 ws3">,<span class="ff1">...</span><span class="ls1">,</span><span class="ff2">v<span class="ff4 fs2 ls30 v1">k</span></span></span><span class="ws1d">um conjunto de veto<span class="_0 blank"></span>res em <span class="ff6 ls18">R<span class="ff4 fs2 ls31 v2">n</span></span><span class="ws5">.<span class="_2 blank"> </span>O conjunto</span></span></span></div><div class="t m0 x7 h4 y1b ff1 fs1 fc1 sc0 ls2f ws5">de to<span class="_4 blank"> </span>das as combinações linea<span class="_0 blank"></span>res de <span class="ff2 ws3">v</span><span class="fs2 ls0 v1">1</span><span class="ff3 ws3">,<span class="ff1">...</span><span class="ls1">,</span><span class="ff2">v<span class="ff4 fs2 ls32 v1">k</span></span></span>:</div><div class="t m0 xb h7 y1c ff5 fs3 fc1 sc0 ls33">L<span class="ff7 ls34">[<span class="ff2 fs1 ls2f ws3">v<span class="ff1 fs2 ls0 v1">1</span><span class="ff3 ws1e">, ..., </span>v<span class="ff4 fs2 ls35 v1">k</span></span><span class="ls36">]</span></span><span class="ls37">\ue011<span class="ls38 v0">f</span><span class="ff4 fs1 lsd">c<span class="ff1 fs2 ls0 v1">1</span><span class="ff2 ls2f ws3">v<span class="ff1 fs2 ls39 v1">1</span></span></span><span class="ff7 lsa">+<span class="ff3 fs1 ls2f ws1f">... </span><span class="ls3a">+<span class="ff4 fs1 lsd">c<span class="fs2 ls32 v1">k</span><span class="ff2 ls2f ws3">v</span><span class="fs2 ls3b v1">k</span><span class="ffa ls3c">:</span><span class="ls8">c<span class="ff1 fs2 ls3d v1">1</span><span class="ff3 ls2f ws6">, ..., </span>c<span class="fs2 ls3e v1">k</span></span></span></span></span><span class="ls3f">2<span class="ff6 fs1 ls40">R</span><span class="ls2f v0">g</span></span></span></div><div class="t m0 x7 h4 y1d ff1 fs1 fc1 sc0 ls2f ws20">é denominado <span class="fc2 ws21">conjunto gerado </span><span class="ws10">p o<span class="_0 blank"></span>r<span class="_3 blank"> </span><span class="ff2 ws3">v</span><span class="fs2 ls0 v1">1</span><span class="ff3 ws3">,<span class="ff1">...</span><span class="ls16">,</span><span class="ff2">v<span class="ff4 fs2 lse v1">k</span></span>.</span></span></div><div class="t m0 x7 h9 y1e ff1 fs1 fc2 sc0 ls2f ws1b">De\u2026nição: <span class="fc1 ws1c">Seja <span class="ff2 ws3">v</span><span class="fs2 ls2 v1">1</span><span class="ff3 ws3">,<span class="ff1">...</span><span class="ls1">,</span><span class="ff2">v<span class="ff4 fs2 ls30 v1">k</span></span></span><span class="ws1d">um conjunto de veto<span class="_0 blank"></span>res em <span class="ff6 ls18">R<span class="ff4 fs2 ls31 v2">n</span></span><span class="ws22">. Um</span></span></span></div><div class="t m0 x7 h9 y1f ff1 fs1 fc1 sc0 ls2f ws23">sub conjunto<span class="_6 blank"> </span><span class="ff4 ls41">V</span><span class="ws24">de <span class="ff6 ls42">R<span class="ff4 fs2 ls43 v2">n</span></span><span class="ws5">tal que <span class="ff4 ls44">V<span class="ff7 fs3 ls45 v0">=<span class="ff5 ls46">L</span><span class="ls47">[</span></span></span><span class="ff2 ws3 v0">v</span><span class="fs2 ls22 v1">1</span><span class="ff3 ws6 v0">, ..., <span class="ff2 ws3">v<span class="ff4 fs2 ls48 v1">k</span><span class="ff7 fs3 ls49">]</span><span class="ff1 ws25">é um <span class="fc2 ws7">sub espaço</span></span></span></span></span></span></div><div class="t m0 x7 hf y20 ff1 fs1 fc2 sc0 ls2f ws26">veto<span class="_0 blank"></span>rial <span class="fc1 ws27 v0">de <span class="ff6 ls4a">R<span class="ff4 fs2 ls25 v2">n</span></span>.</span></div><div class="t m0 x5 h5 y21 ff1 fs2 fc1 sc0 ls2f">4</div></div><div class="c x0 y7 w2 h2"><div class="t m0 x6 h3 y8 ff1 fs0 fc0 sc0 ls2f ws28">Co<span class="_0 blank"></span>nju<span class="_0 blank"></span>nt<span class="_0 blank"></span>os Ge<span class="_0 blank"></span>rad<span class="_0 blank"></span>o<span class="_0 blank"></span>res:<span class="_a blank"> </span>Ge<span class="_0 blank"></span>om<span class="_0 blank"></span>etri<span class="_0 blank"></span>a</div><div class="t m0 x7 h10 y22 ff1 fs1 fc1 sc0 ls2f ws10">Conjunto<span class="_6 blank"> </span>gerado<span class="_6 blank"> </span>p or<span class="_6 blank"> </span>um<span class="_6 blank"> </span>único<span class="_6 blank"> </span>vetor<span class="_6 blank"> </span><span class="ff2 ls4b">v<span class="ff5 fs3 ls3f">2</span><span class="ff6 ls4a">R</span></span><span class="fs2 ls4c v2">2</span><span class="ff3">.</span></div><div class="t m0 x5 h5 y23 ff1 fs2 fc1 sc0 ls2f">5</div></div><div class="c x0 y16 w2 h2"><div class="t m0 x6 h3 y24 ff1 fs0 fc0 sc0 ls2f ws1a">Co<span class="_0 blank"></span>nju<span class="_0 blank"></span>nt<span class="_0 blank"></span>os Ge<span class="_0 blank"></span>rad<span class="_0 blank"></span>o<span class="_0 blank"></span>res:<span class="_a blank"> </span>Ge<span class="_0 blank"></span>om<span class="_0 blank"></span>etri<span class="_0 blank"></span>a</div><div class="t m0 x7 h4 y25 ff1 fs1 fc1 sc0 ls2f ws1">Conjunto gerado por dois veto<span class="_0 blank"></span>res linearmente dependentes</div><div class="t m0 x7 h11 y26 ff2 fs1 fc1 sc0 ls2f ws3">v<span class="ff1 fs2 ls3d v1">1</span><span class="ff3 ls4d">,</span>v<span class="ff1 fs2 ls4e v1">2</span><span class="ff5 fs3 ls4 v0">2<span class="ff6 fs1 ls24">R<span class="ff1 fs2 ls0 v2">2</span><span class="ff3 ls2f">.</span></span></span></div><div class="t m0 x5 h5 y18 ff1 fs2 fc1 sc0 ls2f">6</div></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf3" class="pf w0 h0" data-page-no="3"><div class="pc pc3 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x0 y0 w1 h1" alt="" src="https://files.passeidireto.com/f576e19b-d334-4686-aa50-430d24758f0a/bg3.png"><div class="c x0 y1 w2 h2"><div class="t m0 x6 h3 y19 ff1 fs0 fc0 sc0 ls2f ws28">Co<span class="_0 blank"></span>nju<span class="_0 blank"></span>nt<span class="_0 blank"></span>os Ge<span class="_0 blank"></span>rad<span class="_0 blank"></span>o<span class="_0 blank"></span>res:<span class="_a blank"> </span>Ge<span class="_0 blank"></span>om<span class="_0 blank"></span>etri<span class="_0 blank"></span>a</div><div class="t m0 x7 h4 y27 ff1 fs1 fc1 sc0 ls2f wsd">Conjunto gerado p<span class="_4 blank"> </span>o<span class="_0 blank"></span>r dois veto<span class="_0 blank"></span>res linearmente independentes</div><div class="t m0 x7 h12 y28 ff2 fs1 fc1 sc0 ls2f ws3">v<span class="ff1 fs2 ls3d v1">1</span><span class="ff3 ls4d">,</span>v<span class="ff1 fs2 ls4e v1">2</span><span class="ff5 fs3 ls4 v0">2<span class="ff6 fs1 ls24">R<span class="ff1 fs2 ls0 v2">3</span><span class="ff3 ls2f">.</span></span></span></div><div class="t m0 x5 h5 y29 ff1 fs2 fc1 sc0 ls2f">7</div></div><div class="c x0 y7 w2 h2"><div class="t m0 x6 h3 y8 ff1 fs0 fc0 sc0 ls2f ws1a">Co<span class="_0 blank"></span>nju<span class="_0 blank"></span>nt<span class="_0 blank"></span>os Ge<span class="_0 blank"></span>rad<span class="_0 blank"></span>o<span class="_0 blank"></span>res:<span class="_a blank"> </span>Ge<span class="_0 blank"></span>om<span class="_0 blank"></span>etri<span class="_0 blank"></span>a</div><div class="t m0 x7 h4 y2a ff1 fs1 fc1 sc0 ls2f ws1">Conjunto gerado por dois veto<span class="_0 blank"></span>res linearmente dependentes</div><div class="t m0 x7 h13 y2b ff2 fs1 fc1 sc0 ls2f ws3">v<span class="ff1 fs2 ls3d v1">1</span><span class="ff3 ls4d">,</span>v<span class="ff1 fs2 ls3d v1">2</span><span class="ff3 ls4d">,</span>v<span class="ff1 fs2 ls4e v1">3</span><span class="ff5 fs3 ls4 v0">2<span class="ff6 fs1 ls24">R<span class="ff1 fs2 ls2 v2">3</span><span class="ff3 ls2f">.</span></span></span></div><div class="t m0 x5 h5 y23 ff1 fs2 fc1 sc0 ls2f">8</div></div><div class="c x0 y16 w2 h2"><div class="t m0 x6 h3 y24 ff1 fs0 fc0 sc0 ls2f ws1a">Co<span class="_0 blank"></span>nju<span class="_0 blank"></span>nt<span class="_0 blank"></span>os Ge<span class="_0 blank"></span>rad<span class="_0 blank"></span>o<span class="_0 blank"></span>res:<span class="_a blank"> </span>Ex<span class="_0 blank"></span>em<span class="_0 blank"></span>plo<span class="_0 blank"></span>s</div><div class="t m0 x7 hd y2c ff1 fs1 fc2 sc0 ls2f wsd">Exemplo 1:<span class="_2 blank"> </span><span class="fc1 ws5">Qualquer reta em <span class="ff6 ls5">R<span class="ff4 fs2 ls6 v2">n</span></span><span class="ws29">que passa p<span class="_4 blank"> </span>ela o<span class="_0 blank"></span>rigem é gerada p<span class="_4 blank"> </span>o<span class="_0 blank"></span>r</span></span></div><div class="t m0 x7 h4 y2d ff1 fs1 fc1 sc0 ls2f ws2a">um veto<span class="_0 blank"></span>r não-nulo da reta.</div><div class="t m0 xc h14 y2e ffb fs5 fc0 sc0 ls4f">I<span class="ff1 fs6 fc1 ls50 v1">O<span class="fc2 ls2f ws2b">eixo <span class="ff4 ls51">x</span><span class="fs2 ls52 v1">1</span><span class="fc1 ws2c">é<span class="_6 blank"> </span>gerado<span class="_b blank"> </span>p o<span class="_0 blank"></span>r<span class="_6 blank"> </span><span class="ff2 ws2d">e</span><span class="fs2 ls53 v1">1</span><span class="ff7 fs7 ls54">=<span class="ls55 v0">(</span></span><span class="ls56">1<span class="ff3 ls57">,</span></span><span class="ws2d">0<span class="ff3 ws2e">, ..., </span><span class="ls58">0<span class="ff7 fs7 ls59 v0">)</span><span class="ff3 ls5a">,</span></span><span class="ws2f">p ois<span class="_b blank"> </span>qualquer<span class="_6 blank"> </span>elemento<span class="_b blank"> </span>da</span></span></span></span></span></div><div class="t m0 xd h15 y2f ff1 fs6 fc1 sc0 ls2f ws30">reta <span class="ff4 ls5b">x</span><span class="fs2 ls5c v1">1</span><span class="ws31">p o de<span class="_b blank"> </span>ser<span class="_6 blank"> </span>exp<span class="_0 blank"></span>resso<span class="_6 blank"> </span>como:</span></div><div class="t m0 xe h16 y30 ff4 fs6 fc1 sc0 ls5d">c<span class="ff3 ls5e">.<span class="ff8 ls2f v4">0</span></span></div><div class="t m0 xf h17 y31 ff8 fs6 fc1 sc0 ls2f">B</div><div class="t m0 xf h17 y32 ff8 fs6 fc1 sc0 ls2f">B</div><div class="t m0 xf h17 y33 ff8 fs6 fc1 sc0 ls2f">B</div><div class="t m0 xf h17 y34 ff8 fs6 fc1 sc0 ls2f">@</div><div class="t m0 x10 h15 y35 ff1 fs6 fc1 sc0 ls2f">1</div><div class="t m0 x10 h15 y36 ff1 fs6 fc1 sc0 ls2f">0</div><div class="t m0 x11 h15 y37 ff1 fs6 fc1 sc0 ls2f">.</div><div class="t m0 x11 h15 y38 ff1 fs6 fc1 sc0 ls2f">.</div><div class="t m0 x11 h15 y39 ff1 fs6 fc1 sc0 ls2f">.</div><div class="t m0 x10 h15 y3a ff1 fs6 fc1 sc0 ls2f">0</div><div class="t m0 x12 h17 y3b ff8 fs6 fc1 sc0 ls2f">1</div><div class="t m0 x12 h17 y3c ff8 fs6 fc1 sc0 ls2f">C</div><div class="t m0 x12 h17 y3d ff8 fs6 fc1 sc0 ls2f">C</div><div class="t m0 x12 h17 y3e ff8 fs6 fc1 sc0 ls2f">C</div><div class="t m0 x12 h18 y3f ff8 fs6 fc1 sc0 ls5f">A<span class="ff3 ls60 v5">,<span class="ff1 ls2f ws32">com <span class="ff4 ls61">c<span class="ff5 fs7 ls62">2</span></span><span class="ff6">R</span></span></span></div><div class="t m0 xc h19 y40 ffb fs5 fc0 sc0 ls4f">I<span class="ff1 fs6 fc1 ls63 v1">A<span class="fc2 ls2f ws33">reta diagonal <span class="fc1 ws31">é<span class="_b blank"> </span>gerada<span class="_6 blank"> </span>pelo<span class="_6 blank"> </span>veto<span class="_0 blank"></span>r<span class="_6 blank"> </span><span class="ff7 fs7 ls64 v0">(</span><span class="ws2d">1<span class="ff3 ls65">,</span>1<span class="ff3 ws34">, ..., </span><span class="ls66">1<span class="ff7 fs7 ls64 v0">)</span></span><span class="ws2c">,<span class="_b blank"> </span>p ois<span class="_6 blank"> </span>qualquer<span class="_b blank"> </span>elemento</span></span></span></span></span></div><div class="t m0 xd h15 y41 ff1 fs6 fc1 sc0 ls2f ws35">da diagonal po<span class="_4 blank"> </span>de ser expresso como:</div><div class="t m0 x13 h1a y42 ff4 fs6 fc1 sc0 ls67">c<span class="ff8 ls2f v6">0</span></div><div class="t m0 x14 h17 y43 ff8 fs6 fc1 sc0 ls2f">B</div><div class="t m0 x14 h17 y44 ff8 fs6 fc1 sc0 ls2f">@</div><div class="t m0 x15 h15 y45 ff1 fs6 fc1 sc0 ls2f">1</div><div class="t m0 x10 h15 y46 ff1 fs6 fc1 sc0 ls2f">.</div><div class="t m0 x10 h15 y47 ff1 fs6 fc1 sc0 ls2f">.</div><div class="t m0 x10 h15 y48 ff1 fs6 fc1 sc0 ls2f">.</div><div class="t m0 x15 h15 y49 ff1 fs6 fc1 sc0 ls2f">1</div><div class="t m0 x16 h17 y4a ff8 fs6 fc1 sc0 ls2f">1</div><div class="t m0 x16 h17 y4b ff8 fs6 fc1 sc0 ls2f">C</div><div class="t m0 x16 h1b y4c ff8 fs6 fc1 sc0 ls68">A<span class="ff3 ls69 v7">,<span class="ff1 ls2f ws36">com <span class="ff4 ls61">c<span class="ff5 fs7 ls62">2</span></span><span class="ff6">R</span></span></span></div><div class="t m0 x5 h5 y4d ff1 fs2 fc1 sc0 ls2f">9</div></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf4" class="pf w0 h0" data-page-no="4"><div class="pc pc4 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x0 y0 w1 h1" alt="" src="https://files.passeidireto.com/f576e19b-d334-4686-aa50-430d24758f0a/bg4.png"><div class="c x0 y1 w2 h2"><div class="t m0 x6 h3 y19 ff1 fs0 fc0 sc0 ls2f ws28">Co<span class="_0 blank"></span>nju<span class="_0 blank"></span>nt<span class="_0 blank"></span>os Ge<span class="_0 blank"></span>rad<span class="_0 blank"></span>o<span class="_0 blank"></span>res:<span class="_a blank"> </span>Ex<span class="_0 blank"></span>em<span class="_0 blank"></span>plo<span class="_0 blank"></span>s</div><div class="t m0 x7 h1c y4e ff1 fs1 fc2 sc0 ls2f wsd">Exemplo 2:<span class="_2 blank"> </span><span class="fc1 ls6a">O</span><span class="ws37">plano <span class="ff4 ls6b">x</span><span class="fs2 ls0 v1">1</span><span class="ff4 ls6c">x</span><span class="fs2 ls6d v1">2</span><span class="fc1 ws38">de <span class="ff6 ls6e">R</span><span class="fs2 ls6f v2">3</span><span class="ws39">é<span class="_6 blank"> </span>gerado<span class="_6 blank"> </span>p elos<span class="_3 blank"> </span>veto<span class="_0 blank"></span>res<span class="_3 blank"> </span>unitários</span></span></span></div><div class="t m0 x7 h1d y4f ff4 fs1 fc1 sc0 ls70">e<span class="ff1 fs2 ls71 v1">1</span><span class="ff7 fs3 ls72 v0">=<span class="ls73 v0">(</span><span class="ff1 fs1 ls2f ws3">1<span class="ff3 ls74">,</span>0<span class="ff3 ls74">,</span><span class="ls75">0</span></span><span class="ls76 v0">)</span><span class="ff1 fs1 ls77">e</span></span><span class="v0">e<span class="ff1 fs2 ls78 v1">2</span><span class="ff7 fs3 ls72">=<span class="ls73 v0">(</span></span><span class="ff1 ls2f ws3">0<span class="ff3 ls74">,</span>1<span class="ff3 ls74">,</span><span class="ls79">0<span class="ff7 fs3 ls7a v0">)</span></span><span class="wsd">, p<span class="_4 blank"> </span>ois cada elemento do plano </span></span><span class="ls7b">x<span class="ff1 fs2 ls7c v1">1</span>x<span class="ff1 fs2 ls7d v1">2</span><span class="ff1 ls2f ws7">po de</span></span></span></div><div class="t m0 x7 h4 y50 ff1 fs1 fc1 sc0 ls2f ws2a">ser exp<span class="_0 blank"></span>resso como:</div><div class="t m0 x17 h1e y51 ff4 fs1 fc1 sc0 lsd">c<span class="ff1 fs2 ls7e v1">1</span><span class="ff8 ls2f v8">0</span></div><div class="t m0 x18 h1f y52 ff8 fs1 fc1 sc0 ls2f">@</div><div class="t m0 x19 h4 y53 ff1 fs1 fc1 sc0 ls2f">1</div><div class="t m0 x19 h4 y54 ff1 fs1 fc1 sc0 ls2f">0</div><div class="t m0 x19 h4 y55 ff1 fs1 fc1 sc0 ls2f">0</div><div class="t m0 x1a h1f y56 ff8 fs1 fc1 sc0 ls2f">1</div><div class="t m0 x1a h20 y52 ff8 fs1 fc1 sc0 ls7f">A<span class="ff7 fs3 lsa v9">+</span><span class="ff4 ls8 v9">c</span><span class="ff1 fs2 ls80 va">2</span><span class="ls2f vb">0</span></div><div class="t m0 x1b h1f y57 ff8 fs1 fc1 sc0 ls2f">@</div><div class="t m0 x1c h4 y58 ff1 fs1 fc1 sc0 ls2f">0</div><div class="t m0 x1c h4 y59 ff1 fs1 fc1 sc0 ls2f">1</div><div class="t m0 x1c h4 y5a ff1 fs1 fc1 sc0 ls2f">0</div><div class="t m0 x1d h1f y5b ff8 fs1 fc1 sc0 ls2f">1</div><div class="t m0 x1d h1f y57 ff8 fs1 fc1 sc0 ls81">A<span class="ff3 ls82 v9">,<span class="ff1 ls2f ws3a">com <span class="ff4 lsd">c</span><span class="fs2 ls0 v1">1</span><span class="ff3 ls74">,<span class="ff4 lsd">c</span></span><span class="fs2 ls83 v1">2</span><span class="ff5 fs3 ls84 v0">2</span><span class="ff6 v0">R</span></span></span></div><div class="t m0 x7 h4 y5c ff1 fs1 fc2 sc0 ls2f wsd">Exemplo 3:<span class="_2 blank"> </span><span class="fc1 ls6a">O</span>espaço euclidiano <span class="ff4 fc1 ls85">n</span><span class="fc1">-dimensional é gerado p<span class="_4 blank"> </span>elos veto<span class="_0 blank"></span>res</span></div><div class="t m0 x7 hd y5d ff2 fs1 fc1 sc0 ls86">e<span class="ff1 fs2 ls2 v1">1</span><span class="ff3 ls2f ws1e">, ...,<span class="_6 blank"> </span></span><span class="ls2f ws3">e<span class="ff4 fs2 ls25 v1">n</span><span class="ff1 ws5">, p<span class="_4 blank"> </span>ois qualquer veto<span class="_0 blank"></span>r arbitrário de <span class="ff6 ls5">R<span class="ff4 fs2 ls23 v2">n</span></span><span class="ws7">po de<span class="_6 blank"> </span>ser<span class="_3 blank"> </span>exp<span class="_0 blank"></span>resso</span></span></span></div><div class="t m0 x7 h4 y5e ff1 fs1 fc1 sc0 ls2f">como:</div><div class="t m0 xc h21 y5f ff4 fs1 fc1 sc0 ls8">c<span class="ff1 fs2 ls2f v1">1</span></div><div class="t m0 xd h1f y60 ff8 fs1 fc1 sc0 ls2f">0</div><div class="t m0 xd h1f y61 ff8 fs1 fc1 sc0 ls2f">B</div><div class="t m0 xd h1f y62 ff8 fs1 fc1 sc0 ls2f">B</div><div class="t m0 xd h1f y63 ff8 fs1 fc1 sc0 ls2f">B</div><div class="t m0 xd h1f y64 ff8 fs1 fc1 sc0 ls2f">@</div><div class="t m0 x2 h4 y65 ff1 fs1 fc1 sc0 ls2f">1</div><div class="t m0 x2 h4 y66 ff1 fs1 fc1 sc0 ls2f">0</div><div class="t m0 x2 h4 y67 ff1 fs1 fc1 sc0 ls2f">.</div><div class="t m0 x2 h4 y68 ff1 fs1 fc1 sc0 ls2f">.</div><div class="t m0 x2 h4 y69 ff1 fs1 fc1 sc0 ls2f">.</div><div class="t m0 x2 h4 y6a ff1 fs1 fc1 sc0 ls2f">0</div><div class="t m0 x1e h1f y6b ff8 fs1 fc1 sc0 ls2f">1</div><div class="t m0 x1e h1f y6c ff8 fs1 fc1 sc0 ls2f">C</div><div class="t m0 x1e h1f y6d ff8 fs1 fc1 sc0 ls2f">C</div><div class="t m0 x1e h1f y6e ff8 fs1 fc1 sc0 ls2f">C</div><div class="t m0 x1e h22 y6f ff8 fs1 fc1 sc0 ls87">A<span class="ff7 fs3 lsa vc">+</span><span class="ff4 ls88 vc">c</span><span class="ff1 fs2 ls2f vd">2</span></div><div class="t m0 x1f h1f y70 ff8 fs1 fc1 sc0 ls2f">0</div><div class="t m0 x1f h1f y71 ff8 fs1 fc1 sc0 ls2f">B</div><div class="t m0 x1f h1f y72 ff8 fs1 fc1 sc0 ls2f">B</div><div class="t m0 x1f h1f y73 ff8 fs1 fc1 sc0 ls2f">B</div><div class="t m0 x1f h1f y74 ff8 fs1 fc1 sc0 ls2f">@</div><div class="t m0 x1a h4 y75 ff1 fs1 fc1 sc0 ls2f">0</div><div class="t m0 x1a h4 y76 ff1 fs1 fc1 sc0 ls2f">1</div><div class="t m0 x1a h4 y77 ff1 fs1 fc1 sc0 ls2f">.</div><div class="t m0 x1a h4 y78 ff1 fs1 fc1 sc0 ls2f">.</div><div class="t m0 x1a h4 y79 ff1 fs1 fc1 sc0 ls2f">.</div><div class="t m0 x1a h4 y7a ff1 fs1 fc1 sc0 ls2f">0</div><div class="t m0 x20 h1f y7b ff8 fs1 fc1 sc0 ls2f">1</div><div class="t m0 x20 h1f y7c ff8 fs1 fc1 sc0 ls2f">C</div><div class="t m0 x20 h1f y7d ff8 fs1 fc1 sc0 ls2f">C</div><div class="t m0 x20 h1f y7e ff8 fs1 fc1 sc0 ls2f">C</div><div class="t m0 x20 h23 y7f ff8 fs1 fc1 sc0 ls89">A<span class="ff7 fs3 ls8a vc">+</span><span class="ff3 ls2f ws8 vc">... <span class="ff7 fs3 lsa">+<span class="ff4 fs1 ls70">c<span class="fs2 ls2f v1">n</span></span></span></span></div><div class="t m0 x21 h1f y70 ff8 fs1 fc1 sc0 ls2f">0</div><div class="t m0 x21 h1f y71 ff8 fs1 fc1 sc0 ls2f">B</div><div class="t m0 x21 h1f y72 ff8 fs1 fc1 sc0 ls2f">B</div><div class="t m0 x21 h1f y73 ff8 fs1 fc1 sc0 ls2f">B</div><div class="t m0 x21 h1f y74 ff8 fs1 fc1 sc0 ls2f">@</div><div class="t m0 x22 h4 y80 ff1 fs1 fc1 sc0 ls2f">0</div><div class="t m0 x22 h4 y81 ff1 fs1 fc1 sc0 ls2f">0</div><div class="t m0 x23 h4 y82 ff1 fs1 fc1 sc0 ls2f">.</div><div class="t m0 x23 h4 y83 ff1 fs1 fc1 sc0 ls2f">.</div><div class="t m0 x23 h4 y84 ff1 fs1 fc1 sc0 ls2f">.</div><div class="t m0 x22 h4 y85 ff1 fs1 fc1 sc0 ls2f">1</div><div class="t m0 x24 h1f y86 ff8 fs1 fc1 sc0 ls2f">1</div><div class="t m0 x24 h1f y87 ff8 fs1 fc1 sc0 ls2f">C</div><div class="t m0 x24 h1f y88 ff8 fs1 fc1 sc0 ls2f">C</div><div class="t m0 x24 h1f y89 ff8 fs1 fc1 sc0 ls2f">C</div><div class="t m0 x24 h24 y8a ff8 fs1 fc1 sc0 ls8b">A<span class="ff3 ls82 vc">,<span class="ff1 ls2f ws3b">com <span class="ff4 ls8">c</span><span class="fs2 ls0 v1">1</span><span class="ff3 ws6">, ...,<span class="_c blank"> </span><span class="ff4 ls8c">c<span class="fs2 ls8d v1">n</span><span class="ff5 fs3 ls84 v0">2</span></span><span class="ff6 v0">R</span></span></span></span></div><div class="t m0 x25 h5 y29 ff1 fs2 fc1 sc0 ls2f ws3c">10</div></div><div class="c x0 y7 w2 h2"><div class="t m0 x6 h3 y8 ff1 fs0 fc0 sc0 ls2f ws1a">Co<span class="_0 blank"></span>nju<span class="_0 blank"></span>nt<span class="_0 blank"></span>os Ge<span class="_0 blank"></span>rad<span class="_0 blank"></span>o<span class="_0 blank"></span>res</div><div class="t m0 x7 h25 y8b ff1 fs1 fc2 sc0 ls2f ws14">T<span class="_9 blank"></span>eo<span class="_0 blank"></span>rema: <span class="fc1 ws3d">Seja <span class="ff2 ws3">v</span><span class="fs2 ls0 v1">1</span><span class="ff3 ws3">,<span class="ff1">...</span><span class="ls1">,</span><span class="ff2">v<span class="ff4 fs2 ls17 v1">k</span></span></span><span class="ws2">um conjunto de <span class="ff4 ls1e">k</span><span class="ws3e">vetores em <span class="ff6 ls5">R<span class="ff4 fs2 ls43 v2">n</span></span><span class="ws3f">e considere</span></span></span></span></div><div class="t m0 x7 hd y8c ff1 fs1 fc1 sc0 ls2f ws40">um veto<span class="_0 blank"></span>r qualquer <span class="ff2 ls8e">b<span class="ff5 fs3 ls4">2</span><span class="ff6 ls42">R<span class="ff4 fs2 ls8f v2">n</span></span></span><span class="ws41">.<span class="_2 blank"> </span>Então, <span class="ff2 ls90">b</span><span class="ws42">está no espaço<span class="_3 blank"> </span><span class="ff5 fs3 ls33">L<span class="ff7 ls91">[</span></span><span class="ff2 ws3">v</span><span class="fs2 ls3d v1">1</span><span class="ff3 ws6">, ..., <span class="ff2 ws3">v<span class="ff4 fs2 ls92 v1">k</span><span class="ff7 fs3">]</span></span></span></span></span></div><div class="t m0 x7 h7 y8d ff1 fs1 fc1 sc0 ls2f ws23">gerado<span class="_6 blank"> </span>p or<span class="_6 blank"> </span><span class="ff2 ws3">v</span><span class="fs2 ls2 v1">1</span><span class="ff3 ws3">,<span class="ff1">...</span><span class="ls1">,</span><span class="ff2">v<span class="ff4 fs2 ls30 v1">k</span></span></span><span class="ws2">se, e somente se, o sistema <span class="ff4 ws3">A<span class="ff2 ls93">c<span class="ff7 fs3 ls10">=</span><span class="ls90">b</span></span></span><span class="ws43">tem uma</span></span></div><div class="t m0 x7 h26 y8e ff1 fs1 fc1 sc0 ls2f ws44">solução <span class="ff2 ws3">c<span class="ff3 ls1">,</span></span><span class="ws45">onde <span class="ff4 ls94">A<span class="ff7 fs3 ls95 v0">=</span><span class="ff8 ls96 v3">\ue000</span></span><span class="ff2 ws3 v0">v</span><span class="fs2 ls97 v1">1</span><span class="ff5 fs3 ws12 v0">\ue001 \ue001 \ue001<span class="_7 blank"> </span><span class="ff2 fs1 ws3">v<span class="ff4 fs2 ls98 v1">k</span><span class="ff8 ls99 v3">\ue001</span><span class="ff3 v0">.</span></span></span></span></div><div class="t m0 x25 h5 y8f ff1 fs2 fc1 sc0 ls2f ws3c">11</div></div><div class="c x0 y16 w2 h2"><div class="t m0 x6 h3 y24 ff1 fs0 fc0 sc0 ls2f ws1a">Co<span class="_0 blank"></span>nju<span class="_0 blank"></span>nt<span class="_0 blank"></span>os Ge<span class="_0 blank"></span>rad<span class="_0 blank"></span>o<span class="_0 blank"></span>res</div><div class="t m0 x7 h4 y90 ff1 fs1 fc2 sc0 ls2f ws46">Prova: <span class="fc1 ws2">P<span class="_0 blank"></span>ara demonstra<span class="_0 blank"></span>r o teorema anterio<span class="_0 blank"></span>r, basta nota<span class="_0 blank"></span>r que existe</span></div><div class="t m0 x7 h4 y91 ff1 fs1 fc1 sc0 ls2f ws1">uma equivalência entre as seguintes a\u2026rmações:</div><div class="t m0 x26 h27 y92 ff4 fs6 fc0 sc0 ls9a">i<span class="ff3 ls9b">.<span class="ff2 fc1 ls9c">b<span class="ff5 fs7 ls2f ws47">2<span class="_b blank"> </span>L<span class="ff7 ls9d">[</span></span><span class="ls2f ws2d">v<span class="ff1 fs2 ls2 v1">1</span><span class="ff3 ws34">, ..., </span>v<span class="ff4 fs2 ls9e v9">k</span><span class="ff7 fs7 ls9d">]</span><span class="ff3">.</span></span></span></span></div><div class="t m0 x27 h15 y93 ff4 fs6 fc0 sc0 ls2f ws48">ii <span class="ff3 ls9f">.</span><span class="ff1 fc1 ws49">Existem escalar<span class="_0 blank"></span>es <span class="ff4 lsa0">c</span><span class="fs2 ls0 v1">1</span><span class="ff3 ws2e">, ...,<span class="_6 blank"> </span><span class="ff4 lsa1">c<span class="fs2 lsa2 v1">n</span></span></span><span class="ws4a">tais que:</span></span></div><div class="t m0 x28 h28 y94 ff4 fs6 fc1 sc0 lsa0">c<span class="ff1 fs2 lsa3 v1">1</span><span class="ff8 ls2f v6">0</span></div><div class="t m0 x29 h17 y95 ff8 fs6 fc1 sc0 ls2f">B</div><div class="t m0 x29 h17 y96 ff8 fs6 fc1 sc0 ls2f">@</div><div class="t m0 x3 h29 y97 ff4 fs6 fc1 sc0 ls5b">v<span class="ff1 fs2 ls2f ws4b v1">11</span></div><div class="t m0 x2a h15 y98 ff1 fs6 fc1 sc0 ls2f">.</div><div class="t m0 x2a h15 y99 ff1 fs6 fc1 sc0 ls2f">.</div><div class="t m0 x2a h15 y9a ff1 fs6 fc1 sc0 ls2f">.</div><div class="t m0 x3 h29 y9b ff4 fs6 fc1 sc0 ls5b">v<span class="ff1 fs2 lsa4 v1">1<span class="ff4 ls2f">n</span></span></div><div class="t m0 x2b h17 y9c ff8 fs6 fc1 sc0 ls2f">1</div><div class="t m0 x2b h17 y9d ff8 fs6 fc1 sc0 ls2f">C</div><div class="t m0 x2b h2a y9e ff8 fs6 fc1 sc0 lsa5">A<span class="ff7 fs7 lsa6 v7">+</span><span class="ff3 ls2f ws4c v7">... <span class="ff7 fs7 lsa7">+<span class="ff4 fs6 lsa1">c<span class="fs2 lsa8 v9">k</span></span></span></span><span class="ls2f ve">0</span></div><div class="t m0 x2c h17 y9f ff8 fs6 fc1 sc0 ls2f">B</div><div class="t m0 x2c h17 ya0 ff8 fs6 fc1 sc0 ls2f">@</div><div class="t m0 x2d h29 ya1 ff4 fs6 fc1 sc0 ls2f ws2d">v<span class="fs2 lsa9 v9">k<span class="ff1 ls2f">1</span></span></div><div class="t m0 x2e h15 ya2 ff1 fs6 fc1 sc0 ls2f">.</div><div class="t m0 x2e h15 ya3 ff1 fs6 fc1 sc0 ls2f">.</div><div class="t m0 x2e h15 ya4 ff1 fs6 fc1 sc0 ls2f">.</div><div class="t m0 x2d h29 ya5 ff4 fs6 fc1 sc0 ls5b">v<span class="fs2 ls2f ws4d v9">kn</span></div><div class="t m0 x2f h17 ya6 ff8 fs6 fc1 sc0 ls2f">1</div><div class="t m0 x2f h17 ya7 ff8 fs6 fc1 sc0 ls2f">C</div><div class="t m0 x2f h2b ya8 ff8 fs6 fc1 sc0 lsaa">A<span class="ff7 fs7 lsab v7">=</span><span class="ls2f ve">0</span></div><div class="t m0 x30 h17 ya9 ff8 fs6 fc1 sc0 ls2f">B</div><div class="t m0 x30 h17 yaa ff8 fs6 fc1 sc0 ls2f">@</div><div class="t m0 x31 h29 yab ff4 fs6 fc1 sc0 ls2f ws2d">b<span class="ff1 fs2 v1">1</span></div><div class="t m0 x32 h15 yac ff1 fs6 fc1 sc0 ls2f">.</div><div class="t m0 x32 h15 yad ff1 fs6 fc1 sc0 ls2f">.</div><div class="t m0 x32 h15 yae ff1 fs6 fc1 sc0 ls2f">.</div><div class="t m0 x31 h29 yaf ff4 fs6 fc1 sc0 ls2f ws2d">b<span class="fs2 v1">n</span></div><div class="t m0 x33 h17 yb0 ff8 fs6 fc1 sc0 ls2f">1</div><div class="t m0 x33 h17 yb1 ff8 fs6 fc1 sc0 ls2f">C</div><div class="t m0 x33 h17 yb2 ff8 fs6 fc1 sc0 ls2f">A</div><div class="t m0 x34 h15 yb3 ff4 fs6 fc0 sc0 ls2f ws4e">iii <span class="ff3 lsac">.</span><span class="ff1 fc1 ws4f">O sistema de equações linear<span class="_0 blank"></span>es:</span></div><div class="t m0 x35 h17 yb4 ff8 fs6 fc1 sc0 ls2f">0</div><div class="t m0 x35 h17 yb5 ff8 fs6 fc1 sc0 ls2f">B</div><div class="t m0 x35 h17 yb6 ff8 fs6 fc1 sc0 ls2f">B</div><div class="t m0 x35 h17 yb7 ff8 fs6 fc1 sc0 ls2f">B</div><div class="t m0 x35 h17 yb8 ff8 fs6 fc1 sc0 ls2f">@</div><div class="t m0 x1f h27 yb9 ff4 fs6 fc1 sc0 ls5b">v<span class="ff1 fs2 ls2f ws3c v1">11<span class="_d blank"> </span></span><span class="ls2f ws2d">v<span class="ff1 fs2 ws4b v1">21<span class="_d blank"> </span></span><span class="ff5 fs7 ws50">\ue001 \ue001 \ue001<span class="_7 blank"> </span></span>v<span class="fs2 lsa9 v9">k<span class="ff1 ls2f">1</span></span></span></div><div class="t m0 x1f h27 yba ff4 fs6 fc1 sc0 ls5b">v<span class="ff1 fs2 ls2f ws3c v1">12<span class="_d blank"> </span></span><span class="ls2f ws2d">v<span class="ff1 fs2 ws4b v1">22<span class="_d blank"> </span></span><span class="ff5 fs7 ws50">\ue001 \ue001 \ue001<span class="_7 blank"> </span></span>v<span class="fs2 lsa9 v9">k<span class="ff1 ls2f">2</span></span></span></div><div class="t m0 x19 h15 ybb ff1 fs6 fc1 sc0 ls2f">.</div><div class="t m0 x19 h15 ybc ff1 fs6 fc1 sc0 ls2f">.</div><div class="t m0 x19 h2c ybd ff1 fs6 fc1 sc0 lsad">.<span class="ls2f vf">.</span></div><div class="t m0 x36 h15 ybc ff1 fs6 fc1 sc0 ls2f">.</div><div class="t m0 x36 h2c ybd ff1 fs6 fc1 sc0 lsae">.<span class="lsaf v5">.<span class="v10">.</span></span><span class="lsb0 v7">.</span><span class="ls2f vf">.</span></div><div class="t m0 x37 h15 ybc ff1 fs6 fc1 sc0 ls2f">.</div><div class="t m0 x37 h15 ybd ff1 fs6 fc1 sc0 ls2f">.</div><div class="t m0 x1f h27 ybe ff4 fs6 fc1 sc0 ls5b">v<span class="ff1 fs2 lsa4 v1">1<span class="ff4 lsb1">n</span></span><span class="ls2f ws2d">v<span class="ff1 fs2 lsb2 v1">2<span class="ff4 lsb3">n</span></span><span class="ff5 fs7 ws50">\ue001 \ue001 \ue001<span class="_7 blank"> </span></span></span>v<span class="fs2 ls2f ws51 v9">k n</span></div><div class="t m0 x2c h17 ybf ff8 fs6 fc1 sc0 ls2f">1</div><div class="t m0 x2c h17 yc0 ff8 fs6 fc1 sc0 ls2f">C</div><div class="t m0 x2c h17 yc1 ff8 fs6 fc1 sc0 ls2f">C</div><div class="t m0 x2c h17 yc2 ff8 fs6 fc1 sc0 ls2f">C</div><div class="t m0 x2c h17 yc3 ff8 fs6 fc1 sc0 ls2f">A</div><div class="t m0 x38 h17 yc4 ff8 fs6 fc1 sc0 ls2f">0</div><div class="t m0 x38 h17 yc5 ff8 fs6 fc1 sc0 ls2f">B</div><div class="t m0 x38 h17 yc6 ff8 fs6 fc1 sc0 ls2f">B</div><div class="t m0 x38 h17 yc7 ff8 fs6 fc1 sc0 ls2f">B</div><div class="t m0 x38 h17 yc8 ff8 fs6 fc1 sc0 ls2f">@</div><div class="t m0 x39 h29 yc9 ff4 fs6 fc1 sc0 lsb4">c<span class="ff1 fs2 ls2f v1">1</span></div><div class="t m0 x39 h29 yca ff4 fs6 fc1 sc0 lsb4">c<span class="ff1 fs2 ls2f v1">2</span></div><div class="t m0 x3a h15 ycb ff1 fs6 fc1 sc0 ls2f">.</div><div class="t m0 x3a h15 ycc ff1 fs6 fc1 sc0 ls2f">.</div><div class="t m0 x3a h15 ycd ff1 fs6 fc1 sc0 ls2f">.</div><div class="t m0 x39 h29 yce ff4 fs6 fc1 sc0 lsa1">c<span class="fs2 ls2f v9">k</span></div><div class="t m0 x3b h17 ycf ff8 fs6 fc1 sc0 ls2f">1</div><div class="t m0 x3b h17 yd0 ff8 fs6 fc1 sc0 ls2f">C</div><div class="t m0 x3b h17 yd1 ff8 fs6 fc1 sc0 ls2f">C</div><div class="t m0 x3b h17 yd2 ff8 fs6 fc1 sc0 ls2f">C</div><div class="t m0 x3b h2d yd3 ff8 fs6 fc1 sc0 lsb5">A<span class="ff7 fs7 lsb6 v5">=</span><span class="ls2f v11">0</span></div><div class="t m0 x3c h17 yd4 ff8 fs6 fc1 sc0 ls2f">B</div><div class="t m0 x3c h17 yd5 ff8 fs6 fc1 sc0 ls2f">B</div><div class="t m0 x3c h17 yd6 ff8 fs6 fc1 sc0 ls2f">B</div><div class="t m0 x3c h17 yd7 ff8 fs6 fc1 sc0 ls2f">@</div><div class="t m0 x3d h29 yd8 ff4 fs6 fc1 sc0 ls2f ws2d">b<span class="ff1 fs2 v1">1</span></div><div class="t m0 x3d h29 yd9 ff4 fs6 fc1 sc0 ls2f ws2d">b<span class="ff1 fs2 v1">2</span></div><div class="t m0 x3e h15 yda ff1 fs6 fc1 sc0 ls2f">.</div><div class="t m0 x3e h15 ydb ff1 fs6 fc1 sc0 ls2f">.</div><div class="t m0 x3e h15 ydc ff1 fs6 fc1 sc0 ls2f">.</div><div class="t m0 x3d h29 ydd ff4 fs6 fc1 sc0 ls2f ws2d">b<span class="fs2 v9">k</span></div><div class="t m0 x3f h17 yde ff8 fs6 fc1 sc0 ls2f">1</div><div class="t m0 x3f h17 ydf ff8 fs6 fc1 sc0 ls2f">C</div><div class="t m0 x3f h17 ye0 ff8 fs6 fc1 sc0 ls2f">C</div><div class="t m0 x3f h17 ye1 ff8 fs6 fc1 sc0 ls2f">C</div><div class="t m0 x3f h17 ye2 ff8 fs6 fc1 sc0 ls2f">A</div><div class="t m0 xd h27 ye3 ff1 fs6 fc1 sc0 ls2f ws31">p ossui<span class="_b blank"> </span>uma<span class="_6 blank"> </span>solução<span class="_b blank"> </span><span class="ff2 lsb7">c<span class="ff7 fs7 lsb6">=<span class="lsb8 v0">(</span></span><span class="ff4 lsa1">c</span></span><span class="fs2 ls2 v1">1</span><span class="ff3 lsb9">,<span class="ff4 lsa1">c</span></span><span class="fs2 ls2 v1">2</span><span class="ff3 ws34">, ..., <span class="ff4 lsa0">c<span class="fs2 lsba v9">k</span><span class="ff7 fs7 lsbb v0">)</span></span><span class="lsbc v0">.</span><span class="ffb fs5 v10">\ue004</span></span></div><div class="t m0 x25 h5 ye4 ff1 fs2 fc1 sc0 ls2f ws3c">12</div></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf5" class="pf w0 h0" data-page-no="5"><div class="pc pc5 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x0 y0 w1 h1" alt="" src="https://files.passeidireto.com/f576e19b-d334-4686-aa50-430d24758f0a/bg5.png"><div class="c x0 y1 w2 h2"><div class="t m0 x6 h3 y19 ff1 fs0 fc0 sc0 ls2f ws28">Co<span class="_0 blank"></span>nju<span class="_0 blank"></span>nt<span class="_0 blank"></span>os Ge<span class="_0 blank"></span>rad<span class="_0 blank"></span>o<span class="_0 blank"></span>res</div><div class="t m0 x7 hd ye5 ff1 fs1 fc2 sc0 ls2f ws52">Co<span class="_0 blank"></span>rolário: <span class="fc1 ws53">Seja <span class="ff2 ws3">v</span><span class="fs2 ls2 v1">1</span><span class="ff3 ws3">,<span class="ff1">...</span><span class="lsbd">,</span><span class="ff2">v<span class="ff4 fs2 lsbe v1">k</span></span></span><span class="ws54">um conjunto de <span class="ff4 lsbf">k</span><span class="ws55">vetores em <span class="ff6 ls24">R<span class="ff4 fs2 lsc0 v2">n</span></span><span class="ws56">e considere</span></span></span></span></div><div class="t m0 x7 h2e ye6 ff1 fs1 fc1 sc0 ls2f ws57">a matriz <span class="ff4 lsc1">A<span class="ff7 fs3 ls95 v0">=</span><span class="ff8 ls96 v3">\ue000</span></span><span class="ff2 ws3 v0">v</span><span class="fs2 lsc2 v1">1</span><span class="ff5 fs3 ws12 v0">\ue001 \ue001 \ue001<span class="_7 blank"> </span><span class="ff2 fs1 ws3">v<span class="ff4 fs2 lsc3 v1">k</span><span class="ff8 lsc4 v3">\ue001</span></span></span><span class="ws2 v0">de tamanho <span class="ff4 lsc5">n<span class="ff5 fs3 lsa v0">\ue002</span><span class="lsc6 v0">k</span></span><span class="ws58 v0">.<span class="_2 blank"> </span>Então <span class="ff2 ws3">v</span><span class="fs2 ls0 v1">1</span><span class="ff3 ws3">,<span class="ff1">...</span><span class="ls1">,</span><span class="ff2">v<span class="ff4 fs2 v1">k</span></span></span></span></span></div><div class="t m0 x7 hd ye7 ff1 fs1 fc1 sc0 ls2f ws59">gera <span class="ff6 ls24">R<span class="ff4 fs2 ls23 v2">n</span></span><span class="wsd">se, e somente se, o sistema <span class="ff4 ws3">A<span class="ff2 lsc7">x<span class="ff7 fs3 ls12 v0">=</span><span class="lsc8 v0">b</span></span></span><span class="v0">tem solução para</span></span></div><div class="t m0 x7 hd ye8 ff4 fs1 fc1 sc0 ls2f ws5a">qualquer <span class="ff1 wsd">lado direito <span class="ff2 lsc9">b<span class="ff5 fs3 ls84">2</span><span class="ff6 ls18">R</span></span></span><span class="fs2 ls31 v2">n</span><span class="ff3">.</span></div><div class="t m0 x7 hd ye9 ff1 fs1 fc2 sc0 ls2f ws5b">T<span class="_9 blank"></span>eo<span class="_0 blank"></span>rema: <span class="fc1 ws2">Um conjunto que gera <span class="ff6 ls4a">R<span class="ff4 fs2 ls23 v2">n</span></span><span class="wsd">deve conter p<span class="_4 blank"> </span>elo menos <span class="ff4">n</span></span></span></div><div class="t m0 x7 h4 yea ff1 fs1 fc1 sc0 ls2f ws3">veto<span class="_0 blank"></span>res.</div><div class="t m0 x25 h5 y6 ff1 fs2 fc1 sc0 ls2f ws3c">13</div></div><div class="c x0 y7 w2 h2"><div class="t m0 x6 h3 y8 ff1 fs0 fc0 sc0 ls2f ws1a">Co<span class="_0 blank"></span>nju<span class="_0 blank"></span>nt<span class="_0 blank"></span>os Ge<span class="_0 blank"></span>rad<span class="_0 blank"></span>o<span class="_0 blank"></span>res</div><div class="t m0 x7 h2f yeb ff1 fs1 fc2 sc0 ls2f ws5c">Exemplo: <span class="fc1 ws5d">Demonstre que os veto<span class="_0 blank"></span>res <span class="ff2 ws3">v</span><span class="fs2 ls4e v1">1</span><span class="ff7 fs3 ls95">=</span><span class="ff8 v8">0</span></span></div><div class="t m0 x2e h1f yec ff8 fs1 fc1 sc0 ls2f">@</div><div class="t m0 x40 h4 yed ff1 fs1 fc1 sc0 ls2f">1</div><div class="t m0 x40 h4 yee ff1 fs1 fc1 sc0 ls2f">0</div><div class="t m0 x40 h4 yef ff1 fs1 fc1 sc0 ls2f">0</div><div class="t m0 x41 h1f yf0 ff8 fs1 fc1 sc0 ls2f">1</div><div class="t m0 x41 h30 yf1 ff8 fs1 fc1 sc0 lsca">A<span class="ff1 lscb v9">e<span class="ff2 ls2f ws3">v<span class="ff1 fs2 lscc v1">2</span><span class="ff7 fs3 ls72 v0">=</span><span class="ff8 v8">0</span></span></span></div><div class="t m0 x3f h1f yec ff8 fs1 fc1 sc0 ls2f">@</div><div class="t m0 x42 h4 yed ff1 fs1 fc1 sc0 ls2f">0</div><div class="t m0 x42 h4 yee ff1 fs1 fc1 sc0 ls2f">1</div><div class="t m0 x42 h4 yef ff1 fs1 fc1 sc0 ls2f">1</div><div class="t m0 x43 h1f yf0 ff8 fs1 fc1 sc0 ls2f">1</div><div class="t m0 x43 h1f yf1 ff8 fs1 fc1 sc0 lscd">A<span class="ff1 ls2f v9">não</span></div><div class="t m0 x7 h31 yf2 ff1 fs1 fc1 sc0 ls2f ws5e">geram o <span class="ff6 ls5">R</span><span class="fs2 ls3d v2">3</span><span class="ff3">.</span></div><div class="t m0 xc h32 yf3 ffb fs5 fc0 sc0 ls4f">I<span class="ff1 fs6 fc1 ls2f ws5f v1">Se <span class="ff2 ws2d">v<span class="ff1 fs2 lsce v1">1</span><span class="ff1 lscf">e</span>v<span class="ff1 fs2 lsd0 v1">2</span><span class="ff1 ws60">geram o <span class="ff6 lsd1">R</span><span class="fs2 ls0 v12">3</span><span class="ws61">, então pa<span class="_0 blank"></span>ra qualquer <span class="ff2 ls9c">b<span class="ff5 fs7 ls62">2</span><span class="ff6 lsd1">R</span></span><span class="fs2 lsd0 v12">3</span><span class="ws62">existem <span class="ff4 lsd2">c</span><span class="fs2 lsd0 v1">1</span><span class="lscf">e<span class="ff4 lsa1">c</span></span><span class="fs2 v1">2</span></span></span></span></span></span></div><div class="t m0 xd h15 yf4 ff1 fs6 fc1 sc0 ls2f ws63">tal que:</div><div class="t m0 xd h33 yf5 ff4 fs6 fc1 sc0 lsa1">c<span class="ff1 fs2 lsa3 v1">1</span><span class="ff8 ls2f v13">0</span></div><div class="t m0 x44 h17 yf6 ff8 fs6 fc1 sc0 ls2f">@</div><div class="t m0 x45 h15 yf7 ff1 fs6 fc1 sc0 ls2f">1</div><div class="t m0 x45 h15 yf8 ff1 fs6 fc1 sc0 ls2f">0</div><div class="t m0 x45 h15 yf9 ff1 fs6 fc1 sc0 ls2f">0</div><div class="t m0 x46 h17 yfa ff8 fs6 fc1 sc0 ls2f">1</div><div class="t m0 x46 h34 yfb ff8 fs6 fc1 sc0 lsd3">A<span class="ff7 fs7 lsd4 v9">+</span><span class="ff4 lsa0 v9">c</span><span class="ff1 fs2 lsd5 va">2</span><span class="ls2f v14">0</span></div><div class="t m0 x19 h17 yfc ff8 fs6 fc1 sc0 ls2f">@</div><div class="t m0 x47 h15 yfd ff1 fs6 fc1 sc0 ls2f">0</div><div class="t m0 x47 h15 yfe ff1 fs6 fc1 sc0 ls2f">1</div><div class="t m0 x47 h15 yff ff1 fs6 fc1 sc0 ls2f">1</div><div class="t m0 x36 h17 y100 ff8 fs6 fc1 sc0 ls2f">1</div><div class="t m0 x36 h35 y101 ff8 fs6 fc1 sc0 lsb5">A<span class="ff7 fs7 lsb6 v9">=</span><span class="ls2f v14">0</span></div><div class="t m0 x48 h17 y102 ff8 fs6 fc1 sc0 ls2f">@</div><div class="t m0 x1c h29 y103 ff4 fs6 fc1 sc0 ls2f ws2d">b<span class="ff1 fs2 v1">1</span></div><div class="t m0 x1c h29 y104 ff4 fs6 fc1 sc0 ls2f ws2d">b<span class="ff1 fs2 v1">2</span></div><div class="t m0 x1c h29 y105 ff4 fs6 fc1 sc0 ls2f ws2d">b<span class="ff1 fs2 v1">3</span></div><div class="t m0 x49 h17 y106 ff8 fs6 fc1 sc0 ls2f">1</div><div class="t m0 x49 h35 y107 ff8 fs6 fc1 sc0 lsb5">A<span class="ff5 fs7 lsd6 v9">!</span><span class="ls2f v14">0</span></div><div class="t m0 x4a h17 y108 ff8 fs6 fc1 sc0 ls2f">@</div><div class="t m0 x4b h15 y109 ff1 fs6 fc1 sc0 ls2f ws64">1 0</div><div class="t m0 x4b h15 y10a ff1 fs6 fc1 sc0 ls2f ws64">0 1</div><div class="t m0 x4b h15 y10b ff1 fs6 fc1 sc0 ls2f ws64">0 1</div><div class="t m0 x30 h17 y10c ff8 fs6 fc1 sc0 ls2f">1</div><div class="t m0 x30 h36 y10d ff8 fs6 fc1 sc0 lsd7">A<span class="lsd8 v15">\ue012</span><span class="ff4 lsa1 v2">c</span><span class="ff1 fs2 ls2f v16">1</span></div><div class="t m0 x3d h37 y10e ff4 fs6 fc1 sc0 lsa1">c<span class="ff1 fs2 lsd9 v1">2</span><span class="ff8 lsda v13">\ue013</span><span class="ff7 fs7 lsb6 vd">=</span><span class="ff8 ls2f v17">0</span></div><div class="t m0 x4c h17 y10f ff8 fs6 fc1 sc0 ls2f">@</div><div class="t m0 x4d h29 y110 ff4 fs6 fc1 sc0 ls2f ws2d">b<span class="ff1 fs2 v1">1</span></div><div class="t m0 x4d h29 y111 ff4 fs6 fc1 sc0 ls2f ws2d">b<span class="ff1 fs2 v1">2</span></div><div class="t m0 x4d h29 y112 ff4 fs6 fc1 sc0 ls2f ws2d">b<span class="ff1 fs2 v1">3</span></div><div class="t m0 x4e h17 y113 ff8 fs6 fc1 sc0 ls2f">1</div><div class="t m0 x4e h17 y114 ff8 fs6 fc1 sc0 ls2f">A</div><div class="t m0 xc h38 y115 ffb fs5 fc0 sc0 ls4f">I<span class="ff1 fs6 fc1 ls2f ws63 v1">P<span class="_0 blank"></span>ara demostra<span class="_0 blank"></span>r a proposição, precisamos encontra<span class="_0 blank"></span>r <span class="ff4 ws2d">b<span class="ff1 fs2 ls0 v1">1</span><span class="ff1 ls5a">,</span>b<span class="ff1 fs2 ls2 v1">2</span><span class="ff3 lsdb">,</span>b<span class="ff1 fs2 ls5c v1">3</span><span class="ff1 ws4a">tais que</span></span></span></div><div class="t m0 xd h15 y116 ff1 fs6 fc1 sc0 ls2f ws61">o sistema acima não p<span class="_4 blank"> </span>ossua solução.</div><div class="t m0 x25 h5 y117 ff1 fs2 fc1 sc0 ls2f ws3c">14</div></div><div class="c x0 y16 w2 h2"><div class="t m0 x6 h3 y24 ff1 fs0 fc0 sc0 ls2f ws1a">Co<span class="_0 blank"></span>nju<span class="_0 blank"></span>nt<span class="_0 blank"></span>os Ge<span class="_0 blank"></span>rad<span class="_0 blank"></span>o<span class="_0 blank"></span>res</div><div class="t m0 x7 h4 y118 ff1 fs1 fc1 sc0 ls2f ws3">(Cont.)</div><div class="t m0 xc h39 y119 ffb fs5 fc0 sc0 ls4f">I<span class="ff1 fs6 fc1 ls2f ws61 v1">Escalonando a matriz de co<span class="_4 blank"> </span>e\u2026cientes aumentada, obtemos:</span></div><div class="t m0 xa h17 y11a ff8 fs6 fc1 sc0 ls2f">0</div><div class="t m0 xa h17 y11b ff8 fs6 fc1 sc0 ls2f">@</div><div class="t m0 x8 h15 y11c ff1 fs6 fc1 sc0 ls2f ws65">1 0 <span class="ff4 ws2d">b</span><span class="fs2 v1">1</span></div><div class="t m0 x8 h15 y11d ff1 fs6 fc1 sc0 ls2f ws65">0 1 <span class="ff4 ws2d">b</span><span class="fs2 v1">2</span></div><div class="t m0 x8 h15 y11e ff1 fs6 fc1 sc0 ls2f ws65">0 1 <span class="ff4 ws2d">b</span><span class="fs2 v1">3</span></div><div class="t m0 x15 h17 y11f ff8 fs6 fc1 sc0 ls2f">1</div><div class="t m0 x15 h3a y120 ff8 fs6 fc1 sc0 lsdc">A<span class="ff5 fs7 lsdd v9">!</span><span class="ls2f v14">0</span></div><div class="t m0 x2c h17 y121 ff8 fs6 fc1 sc0 ls2f">@</div><div class="t m0 x2d h15 y122 ff1 fs6 fc1 sc0 ls2f ws64">1 0<span class="_e blank"> </span><span class="ff4 ws2d">b</span><span class="fs2 v1">1</span></div><div class="t m0 x2d h15 y123 ff1 fs6 fc1 sc0 ls2f ws64">0 1<span class="_e blank"> </span><span class="ff4 ws2d">b</span><span class="fs2 v1">2</span></div><div class="t m0 x2d h27 y124 ff1 fs6 fc1 sc0 ls2f ws66">0 0 <span class="ff4 ws2d">b</span><span class="fs2 lsde v1">3</span><span class="ff5 fs7 lsa6">\ue000</span><span class="ff4 ws2d">b</span><span class="fs2 v1">2</span></div><div class="t m0 x3e h17 y125 ff8 fs6 fc1 sc0 ls2f">1</div><div class="t m0 x3e h17 y126 ff8 fs6 fc1 sc0 ls2f">A</div><div class="t m0 xd h27 y127 ff1 fs6 fc1 sc0 ls2f ws61">Note que se <span class="ff4 ws2d">b</span><span class="fs2 lsde v1">3</span><span class="ff5 fs7 lsa6">\ue000</span><span class="ff4 ws2d">b</span><span class="fs2 ls53 v1">2</span><span class="ff5 fs7 lsdf">6<span class="ff7 lse0">=</span></span>0, o sistema não p<span class="_4 blank"> </span>ossui solução.</div><div class="t m0 xc h3b y128 ffb fs5 fc0 sc0 ls4f">I<span class="ff1 fs6 fc1 ls2f ws67 v1">Assim, qualquer <span class="ff2 lse1">b<span class="ff5 fs7 ls62">2</span><span class="ff6 lsd1">R<span class="ff1 fs2 ls0 v12">3</span></span></span><span class="ws61">, com <span class="ff4 ws2d">b</span><span class="fs2 lsde v1">3</span><span class="ff5 fs7 lsa6">\ue000</span><span class="ff4 ws2d">b</span><span class="fs2 lse2 v1">2</span><span class="ff5 fs7 lsdf">6<span class="ff7 lse0">=</span></span>0, não é gerado p<span class="_4 blank"> </span>elos veto<span class="_0 blank"></span>res</span></span></div><div class="t m0 xd h15 y129 ff2 fs6 fc1 sc0 ls2f ws2d">v<span class="ff1 fs2 lse3 v1">1</span><span class="ff1 lse4">e</span>v<span class="ff1 fs2 ls0 v1">2</span><span class="ff1">.</span></div><div class="t m0 xc h3c y12a ffb fs5 fc0 sc0 ls4f">I<span class="ff1 fs6 fc1 ls2f ws68 v1">P<span class="_0 blank"></span>or exemplo, <span class="ff2 lse5">w<span class="ff7 fs7 ls54">=</span><span class="ff8 ls2f v13">0</span></span></span></div><div class="t m0 x47 h17 y12b ff8 fs6 fc1 sc0 ls2f">@</div><div class="t m0 x4f h15 y12c ff1 fs6 fc1 sc0 ls2f">1</div><div class="t m0 x4f h15 y12d ff1 fs6 fc1 sc0 ls2f">2</div><div class="t m0 x4f h15 y12e ff1 fs6 fc1 sc0 ls2f">3</div><div class="t m0 x14 h17 y12f ff8 fs6 fc1 sc0 ls2f">1</div><div class="t m0 x14 h17 y130 ff8 fs6 fc1 sc0 lse6">A<span class="ff1 ls2f ws69 v9">não é gerado p<span class="_4 blank"> </span>elos veto<span class="_0 blank"></span>res <span class="ff2 ws2d">v<span class="ff1 fs2 lsd0 v1">1</span><span class="ff1 lscf">e</span>v<span class="ff1 fs2 ls0 v1">2</span></span>, mas</span></div><div class="t m0 xd h3d y131 ff2 fs6 fc1 sc0 lse7">x<span class="ff7 fs7 lse8">=</span><span class="ff8 ls2f v13">0</span></div><div class="t m0 x50 h17 y132 ff8 fs6 fc1 sc0 ls2f">@</div><div class="t m0 x51 h15 y133 ff1 fs6 fc1 sc0 ls2f">1</div><div class="t m0 x51 h15 y134 ff1 fs6 fc1 sc0 ls2f">3</div><div class="t m0 x51 h15 y135 ff1 fs6 fc1 sc0 ls2f">3</div><div class="t m0 x52 h17 y136 ff8 fs6 fc1 sc0 ls2f">1</div><div class="t m0 x52 h17 y137 ff8 fs6 fc1 sc0 lse9">A<span class="ff1 ls2f ws6a v9">é<span class="_b blank"> </span>gerado<span class="_6 blank"> </span>p o<span class="_0 blank"></span>r<span class="_6 blank"> </span><span class="ff2 ws2d">v<span class="ff1 fs2 lsea v1">1</span><span class="ff1 lscf">e</span>v<span class="ff1 fs2 ls0 v1">2</span><span class="ff1">.</span></span></span></div><div class="t m0 x25 h5 y138 ff1 fs2 fc1 sc0 ls2f ws3c">15</div></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf6" class="pf w0 h0" data-page-no="6"><div class="pc pc6 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x0 y0 w1 h1" alt="" src="https://files.passeidireto.com/f576e19b-d334-4686-aa50-430d24758f0a/bg6.png"><div class="c x0 y1 w2 h2"><div class="t m0 x35 he y139 ff1 fs4 fc1 sc0 ls2f ws6b">B<span class="_9 blank"></span>a<span class="_0 blank"></span>se e D<span class="_9 blank"></span>im<span class="_9 blank"></span>e<span class="_0 blank"></span>n<span class="_9 blank"></span>sã<span class="_9 blank"></span>o n<span class="_0 blank"></span>o R<span class="_0 blank"></span>n</div><div class="t m0 x25 h5 y29 ff1 fs2 fc1 sc0 ls2f ws3c">16</div></div><div class="c x0 y7 w2 h2"><div class="t m0 x6 h3 y8 ff1 fs0 fc0 sc0 ls2f ws6c">Ba<span class="_0 blank"></span>se e<span class="_2 blank"> </span>D<span class="_0 blank"></span>im<span class="_0 blank"></span>ens<span class="_0 blank"></span>ão em Rn</div><div class="t m0 x7 hd y13a ff1 fs1 fc2 sc0 ls2f ws1b">De\u2026nição: <span class="fc1 ws6d">Seja <span class="ff2 ws3">v</span><span class="fs2 ls2 v1">1</span><span class="ff3 ws3">,<span class="ff1">...</span><span class="ls1">,</span><span class="ff2">v<span class="ff4 fs2 lseb v1">m</span></span></span><span class="wsd">um conjunto de <span class="ff4 lsec">m</span><span class="ws6e">veto<span class="_0 blank"></span>res em <span class="ff6 ls42">R<span class="ff4 fs2 ls8f v2">n</span></span><span class="ws22">. Dizemos</span></span></span></span></div><div class="t m0 x7 h3e y13b ff1 fs1 fc1 sc0 ls2f ws6f">que <span class="ff2 ws3">v</span><span class="fs2 ls2 v1">1</span><span class="ff3 ws3">,<span class="ff1">...</span><span class="lsed">,</span><span class="ff2">v<span class="ff4 fs2 lseb v1">m</span></span></span><span class="ws29">é uma <span class="fc2 ws70 v0">base <span class="fc1 ws10">de<span class="_6 blank"> </span>um<span class="_6 blank"> </span>sub espaço<span class="_3 blank"> </span><span class="ff4 lsee">V<span class="ff5 fs3 ls12">\ue01a</span><span class="ff6 ls24">R</span><span class="fs2 ls23 v2">n</span></span><span class="ws3">se:</span></span></span></span></div><div class="t m0 x26 h15 y13c ff4 fs6 fc0 sc0 ls9a">i<span class="ff3 ls9b">.<span class="ff2 fc1 ls2f ws2d">v<span class="ff1 fs2 ls2 v1">1</span><span class="ff3">,<span class="ff1">...</span><span class="ls5a">,</span></span><span class="ls56">v<span class="ff4 fs2 lsef v1">m</span></span><span class="ff1 ws71">gera <span class="ff4 lsf0">V</span><span class="ff3">;</span></span></span></span></div><div class="t m0 x27 h15 y13d ff4 fs6 fc0 sc0 ls2f ws48">ii <span class="ff3 ls9f">.</span><span class="ff2 fc1 ws2d">v<span class="ff1 fs2 ls2 v1">1</span><span class="ff3">,<span class="ff1">...</span><span class="ls5a">,</span></span><span class="ls56">v<span class="ff4 fs2 lsef v1">m</span></span><span class="ff1 ws2c">são<span class="_b blank"> </span>linearmente<span class="_b blank"> </span>indep endentes.</span></span></div><div class="t m0 x7 h4 y13e ff1 fs1 fc2 sc0 ls2f ws72">Exemplos: <span class="fc1 ws5">Bases canônicas:</span></div><div class="t m0 x26 h3f y13f ff4 fs6 fc0 sc0 ls9a">i<span class="ff3 lsf1">.<span class="ff5 fs7 fc1 lsf2 v0">f<span class="ff7 ls64 v0">(<span class="ff1 fs6 ls2f ws2d v0">1<span class="ff3 ls65">,</span><span class="ls58">0</span></span><span class="lsbb">)</span></span></span><span class="fc1 lsf3 v0">,<span class="ff7 fs7 ls64 v0">(</span><span class="ff1 ls2f ws2d">0</span><span class="lsf4">,<span class="ff1 ls66">1<span class="ff7 fs7 lsf5 v0">)<span class="ff5 lsf6 v0">g</span></span><span class="ls2f ws35 v0">é uma base do <span class="ff6 lsf7">R</span><span class="fs2 ls2 v12">2</span><span class="ff3">.</span></span></span></span></span></span></div><div class="t m0 x27 h40 y140 ff4 fs6 fc0 sc0 ls2f ws48">ii <span class="ff3 lsf8">.<span class="ff5 fs7 fc1 lsf2 v0">f<span class="ff7 ls64 v0">(</span></span></span><span class="ff1 fc1 ws2d v0">1<span class="ff3 ls65">,</span>0<span class="ff3 ls65">,</span><span class="ls58">0<span class="ff7 fs7 ls59 v0">)</span><span class="ff3 lsf9">,<span class="ff7 fs7 ls64 v0">(</span></span></span>0<span class="ff3 lsb9">,</span>1<span class="ff3 ls65">,</span><span class="ls58">0<span class="ff7 fs7 ls59 v0">)</span><span class="ff3 lsf9">,<span class="ff7 fs7 ls64 v0">(</span></span></span>0<span class="ff3 lsb9">,</span>0<span class="ff3 lsb9">,</span><span class="ls66">1<span class="ff7 fs7 lsf5 v0">)<span class="ff5 lsf6 v0">g</span></span></span><span class="ws35">é uma base do <span class="ff6 lsd1">R</span><span class="fs2 ls0 v12">3</span><span class="ff3">.</span></span></span></div><div class="t m0 x25 h5 y141 ff1 fs2 fc1 sc0 ls2f ws3c">17</div></div><div class="c x0 y16 w2 h2"><div class="t m0 x6 h3 y24 ff1 fs0 fc0 sc0 ls2f ws6c">Ba<span class="_0 blank"></span>se e<span class="_2 blank"> </span>D<span class="_0 blank"></span>im<span class="_0 blank"></span>ens<span class="_0 blank"></span>ão em Rn</div><div class="t m0 x7 hd y142 ff1 fs1 fc2 sc0 ls2f ws14">T<span class="_9 blank"></span>eo<span class="_0 blank"></span>rema: <span class="fc1 ws73">Qualquer base de <span class="ff6 ls24">R<span class="ff4 fs2 ls23 v2">n</span></span><span class="ws74">contém <span class="ff4 lsfa">n</span><span class="ws3">vetores.</span></span></span></div><div class="t m0 xc h38 y143 ffb fs5 fc0 sc0 ls4f">I<span class="ff1 fs6 fc2 ls2f ws75 v1">Prova: <span class="fc1 ws76">Utilizando os resultados derivados anterio<span class="_0 blank"></span>rmente, p<span class="_4 blank"> </span>o<span class="_4 blank"> </span>demos</span></span></div><div class="t m0 xd h15 y144 ff1 fs6 fc1 sc0 ls2f ws77">mostra<span class="_0 blank"></span>r que:</div><div class="t m0 x44 h41 y145 ff4 fs8 fc0 sc0 lsfb">i<span class="ffc lsfc">.<span class="ff1 fc1 ls2f ws78">Um<span class="_4 blank"> </span>a<span class="_b blank"> </span>bas e<span class="_6 blank"> </span>de<span class="_b blank"> </span><span class="ff6 lsfd">R<span class="ff4 fs9 lsfe v18">n</span></span><span class="ws79">nã o<span class="_6 blank"> </span>p o<span class="_4 blank"> </span>de<span class="_b blank"> </span>c o n t e r<span class="_b blank"> </span><span class="fc2">m<span class="_4 blank"> </span>ais<span class="_b blank"> </span></span><span class="ws7a">d o<span class="_6 blank"> </span>que<span class="_b blank"> </span><span class="ff4 lsff">n</span><span class="ws7b">e le m<span class="_4 blank"> </span>ent o s ,<span class="_b blank"> </span>p<span class="_4 blank"> </span>oi s<span class="_b blank"> </span>c a s o</span></span></span></span></span></div><div class="t m0 x53 h42 y146 ff1 fs8 fc1 sc0 ls2f ws79">con t r á ri o<span class="_6 blank"> </span>o<span class="_b blank"> </span>con ju n t o<span class="_6 blank"> </span>em<span class="_b blank"> </span>qu e s t ã o<span class="_6 blank"> </span>não<span class="_b blank"> </span>p<span class="_4 blank"> </span>o<span class="_4 blank"> </span>der ia<span class="_6 blank"> </span>ser<span class="_b blank"> </span>li n earm<span class="_4 blank"> </span>ente</div><div class="t m0 x53 h42 y147 ff1 fs8 fc1 sc0 ls2f ws7b">ind e p<span class="_4 blank"> </span>en d e n t e .</div><div class="t m0 x54 h41 y148 ff4 fs8 fc0 sc0 ls2f ws7c">ii <span class="ffc ls100">.</span><span class="ff1 fc1 ws7d">Um<span class="_4 blank"> </span>a<span class="_b blank"> </span>bas e<span class="_b blank"> </span>d e<span class="_b blank"> </span><span class="ff6 ls101">R<span class="ff4 fs9 ls102 v18">n</span></span><span class="ws79">n ã o<span class="_b blank"> </span>p<span class="_4 blank"> </span>o<span class="_4 blank"> </span>de<span class="_b blank"> </span>co n t e r<span class="_b blank"> </span><span class="fc2 ws7b">m e n o s<span class="_b blank"> </span></span>d o<span class="_b blank"> </span>q u e<span class="_b blank"> </span><span class="ff4 ls103">n</span><span class="ws7b">e le m<span class="_4 blank"> </span>ent o s,<span class="_b blank"> </span>p<span class="_4 blank"> </span>oi s<span class="_b blank"> </span>c a s o</span></span></span></div><div class="t m0 x53 h41 y149 ff1 fs8 fc1 sc0 ls2f ws79">con t r á ri o<span class="_6 blank"> </span>o<span class="_b blank"> </span>con ju n t o<span class="_6 blank"> </span>em<span class="_b blank"> </span>qu e s t ã o<span class="_6 blank"> </span>não<span class="_b blank"> </span>g e raria<span class="_6 blank"> </span><span class="ff6 ls104">R<span class="ff4 fs9 ls105 v18">n</span></span><span class="ffc v0">.</span></div><div class="t m0 xc h43 y14a ffb fs5 fc0 sc0 ls4f">I<span class="ff1 fs6 fc1 ls2f ws61 v1">P<span class="_0 blank"></span>ortanto, uma base de <span class="ff6 ls106">R<span class="ff4 fs2 ls107 v12">n</span></span>deve conter <span class="fc2 ws7e">exatamente </span><span class="ff4 ls108">n</span><span class="ws7f">veto<span class="_0 blank"></span>res. <span class="ffb fsa v19">\ue004</span></span></span></div><div class="t m0 x25 h5 y14b ff1 fs2 fc1 sc0 ls2f ws3c">18</div></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf7" class="pf w0 h0" data-page-no="7"><div class="pc pc7 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x0 y0 w1 h1" alt="" src="https://files.passeidireto.com/f576e19b-d334-4686-aa50-430d24758f0a/bg7.png"><div class="c x0 y1 w2 h2"><div class="t m0 x6 h3 y19 ff1 fs0 fc0 sc0 ls2f ws80">Ba<span class="_0 blank"></span>se e D<span class="_0 blank"></span>ime<span class="_0 blank"></span>nsã<span class="_0 blank"></span>o em R<span class="_0 blank"></span>n</div><div class="t m0 x7 h9 y14c ff1 fs1 fc2 sc0 ls2f ws5b">T<span class="_9 blank"></span>eo<span class="_0 blank"></span>rema: <span class="fc1 ws81">Seja <span class="ff2 ws3">v</span><span class="fs2 ls0 v1">1</span><span class="ff3 ws3">,<span class="ff1">...</span><span class="ls1">,</span><span class="ff2">v<span class="ff4 fs2 ls23 v1">n</span></span></span><span class="wsd">um conjunto de <span class="ff4 ls109">n</span><span class="ws3e">vetores em <span class="ff6 ls24">R<span class="ff4 fs2 ls23 v2">n</span></span>e</span></span></span></div><div class="t m0 x7 h1d y14d ff4 fs1 fc1 sc0 ls10a">A<span class="ff7 fs3 ls2f ws82 v0">= (<span class="_f blank"> </span></span><span class="ff2 ls2f ws3 v0">v<span class="ff1 fs2 ls10b v1">1</span><span class="ff3 ws83">... </span>v</span><span class="fs2 ls29 v1">n</span><span class="ff7 fs3 ls73 v0">)<span class="ff3 fs1 ls10c">.<span class="ff1 ls2f ws2">Então, as seguintes a\u2026rmações são equivalentes:</span></span></span></div><div class="t m0 x26 h15 y14e ff4 fs6 fc0 sc0 ls9a">i<span class="ff3 ls9b">.<span class="ff2 fc1 ls2f ws2d">v<span class="ff1 fs2 ls2 v19">1</span><span class="ff3">,<span class="ff1">...</span><span class="ls5a">,</span></span><span class="ls56">v<span class="ff4 fs2 lsa2 v1">n</span></span><span class="ff1 ws61">são linea<span class="_0 blank"></span>rmente indep<span class="_4 blank"> </span>endentes;</span></span></span></div><div class="t m0 x27 h44 y14f ff4 fs6 fc0 sc0 ls2f ws48">ii <span class="ff3 ls9f">.</span><span class="ff2 fc1 ws2d">v<span class="ff1 fs2 ls2 v19">1</span><span class="ff3">,<span class="ff1">...</span><span class="ls5a">,</span></span><span class="ls56">v<span class="ff4 fs2 lsa2 v1">n</span></span><span class="ff1 ws84">gera <span class="ff6 lsf7">R<span class="ff4 fs2 ls31 v12">n</span></span>;</span></span></div><div class="t m0 x34 h45 y150 ff4 fs6 fc0 sc0 ls2f ws4e">iii <span class="ff3 lsac">.</span><span class="ff2 fc1 ws2d">v<span class="ff1 fs2 ls2 v19">1</span><span class="ff3">,<span class="ff1">...</span><span class="ls5a">,</span></span><span class="ls56">v<span class="ff4 fs2 lsa2 v1">n</span></span><span class="ff1 ws85">constitui uma base de <span class="ff6 ls106">R<span class="ff4 fs2 ls31 v12">n</span></span>;</span></span></div><div class="t m0 x34 h15 y151 ff4 fs6 fc0 sc0 ls2f ws86">iv <span class="ff3 ls10d">.</span><span class="ff1 fc1 ws4f">O determinante de <span class="ff4 ls10e">A</span><span class="ws35">é não-nulo.</span></span></div><div class="t m0 x25 h5 y6 ff1 fs2 fc1 sc0 ls2f ws3c">19</div></div><div class="c x0 y7 w2 h2"><div class="t m0 x6 h3 y8 ff1 fs0 fc0 sc0 ls2f ws6c">Ba<span class="_0 blank"></span>se e<span class="_2 blank"> </span>D<span class="_0 blank"></span>im<span class="_0 blank"></span>ens<span class="_0 blank"></span>ão em Rn</div><div class="t m0 x7 h9 y152 ff1 fs1 fc1 sc0 ls2f ws5">Intuitivamente, dizemos que o <span class="ff6 ls5">R<span class="ff4 fs2 ls23 v2">n</span></span><span class="ls10f">é</span><span class="fc2 ws3">n-dimensional</span><span class="ws10">,<span class="_6 blank"> </span>p ois<span class="_6 blank"> </span>o<span class="_3 blank"> </span>fato<span class="_6 blank"> </span>de</span></div><div class="t m0 x7 h9 y153 ff1 fs1 fc1 sc0 ls2f ws87">cada base de <span class="ff6 ls4a">R<span class="ff4 fs2 ls6 v2">n</span></span><span class="ws42">conter exatamente <span class="ff4 ls109">n</span><span class="ws2">veto<span class="_0 blank"></span>res nos diz que existem <span class="ff4">n</span></span></span></div><div class="t m0 x7 h9 y154 ff1 fs1 fc1 sc0 ls2f ws88">"direções"<span class="_6 blank"> </span>indep endentes<span class="_6 blank"> </span>em<span class="_3 blank"> </span><span class="ff6 ls5">R<span class="ff4 fs2 ls110 v2">n</span></span><span class="ff3">.</span></div><div class="t m0 x25 h5 y155 ff1 fs2 fc1 sc0 ls2f ws3c">20</div></div><div class="c x0 y16 w2 h2"><div class="t m0 x6 h3 y24 ff1 fs0 fc0 sc0 ls2f ws4">Ex<span class="_0 blank"></span>erc<span class="_0 blank"></span>ício<span class="_0 blank"></span>s</div><div class="t m0 x7 h7 y156 ff1 fs1 fc2 sc0 ls2f wsd">Exercício 1:<span class="_2 blank"> </span><span class="fc1 ws89">O veto<span class="_0 blank"></span>r <span class="ff2 ls111">w<span class="ff7 fs3 ls37">=<span class="ls112 v0">(</span></span></span><span class="ws3">1<span class="ff3 ls113">,<span class="ff5 fs3 ls114">\ue000</span></span>1<span class="ff3 ls115">,</span><span class="ls116">2<span class="ff7 fs3 ls117 v0">)</span></span><span class="ws2">p<span class="_4 blank"> </span>ertence ao conjunto gerado</span></span></span></div><div class="t m0 x7 h1d y157 ff1 fs1 fc1 sc0 ls2f ws23">p elos<span class="_6 blank"> </span>vetores<span class="_6 blank"> </span><span class="ff2 ls118">u<span class="ff7 fs3 ls37 v0">=<span class="ls112 v0">(</span></span></span><span class="ws3 v0">1<span class="ff3 ls119">,</span>2<span class="ff3 ls74">,</span><span class="ls79">3<span class="ff7 fs3 ls117 v0">)</span><span class="ls11a">e<span class="ff2 ls11b">v<span class="ff7 fs3 ls72">=<span class="ls7a v0">(</span></span></span></span></span>3<span class="ff3 ls119">,</span>2<span class="ff3 ls74">,</span><span class="ls79">1<span class="ff7 fs3 ls11c v0">)</span></span><span class="ffa">?</span></span></div><div class="t m0 xc h46 y158 ffb fs5 fc0 sc0 ls4f">I<span class="ff1 fs6 fc1 ls2f ws5f v1">Se <span class="ff2 ls11d">w<span class="ff5 fs7 ls2f ws8a">2<span class="_b blank"> </span>L<span class="ff7 ls11e">[</span></span><span class="ls2f ws2d">u<span class="ff3 lsf4">,</span><span class="ls11f">v<span class="ff7 fs7 ls120">]</span></span><span class="ff1 ws8b">, então devem existir <span class="ff4 lsa1">c</span><span class="fs2 lsd0 v19">1</span><span class="lscf">e<span class="ff4 lsa1">c</span><span class="fs2 lsd0 v19">2</span></span><span class="ws35">tais que:</span></span></span></span></span></div><div class="t m0 x55 h17 y159 ff8 fs6 fc1 sc0 ls2f">0</div><div class="t m0 x55 h17 y15a ff8 fs6 fc1 sc0 ls2f">@</div><div class="t m0 x56 h15 y15b ff1 fs6 fc1 sc0 ls2f">1</div><div class="t m0 x2a h27 y15c ff5 fs7 fc1 sc0 ls121">\ue000<span class="ff1 fs6 ls2f">1</span></div><div class="t m0 x56 h15 y15d ff1 fs6 fc1 sc0 ls2f">2</div><div class="t m0 x57 h17 y15e ff8 fs6 fc1 sc0 ls2f">1</div><div class="t m0 x57 h34 y15f ff8 fs6 fc1 sc0 lsb5">A<span class="ff7 fs7 lse0 v9">=</span><span class="ff4 lsa0 v9">c</span><span class="ff1 fs2 ls122 va">1</span><span class="ls2f v14">0</span></div><div class="t m0 x1d h17 y160 ff8 fs6 fc1 sc0 ls2f">@</div><div class="t m0 x58 h15 y161 ff1 fs6 fc1 sc0 ls2f">1</div><div class="t m0 x58 h15 y162 ff1 fs6 fc1 sc0 ls2f">2</div><div class="t m0 x58 h15 y163 ff1 fs6 fc1 sc0 ls2f">3</div><div class="t m0 x23 h17 y164 ff8 fs6 fc1 sc0 ls2f">1</div><div class="t m0 x23 h34 y165 ff8 fs6 fc1 sc0 lsa5">A<span class="ff7 fs7 ls123 v9">+</span><span class="ff4 lsd2 v9">c</span><span class="ff1 fs2 lsd5 va">2</span><span class="ls2f v14">0</span></div><div class="t m0 x3b h17 y166 ff8 fs6 fc1 sc0 ls2f">@</div><div class="t m0 x59 h15 y167 ff1 fs6 fc1 sc0 ls2f">3</div><div class="t m0 x59 h15 y168 ff1 fs6 fc1 sc0 ls2f">2</div><div class="t m0 x59 h15 y169 ff1 fs6 fc1 sc0 ls2f">1</div><div class="t m0 x5a h17 y16a ff8 fs6 fc1 sc0 ls2f">1</div><div class="t m0 x5a h17 y16b ff8 fs6 fc1 sc0 ls124">A<span class="ff3 ls2f v9">,</span></div><div class="t m0 xd h15 y16c ff1 fs6 fc1 sc0 ls2f ws61">i.e.<span class="_10 blank"> </span>o sistema:<span class="_e blank"> </span><span class="ff8 v1">0</span></div><div class="t m0 x47 h17 y16d ff8 fs6 fc1 sc0 ls2f">@</div><div class="t m0 x4f h15 y16e ff1 fs6 fc1 sc0 ls2f ws8c">1 3</div><div class="t m0 x4f h15 y16f ff1 fs6 fc1 sc0 ls2f ws8c">2 2</div><div class="t m0 x4f h15 y170 ff1 fs6 fc1 sc0 ls2f ws8c">3 1</div><div class="t m0 x10 h17 y171 ff8 fs6 fc1 sc0 ls2f">1</div><div class="t m0 x10 h36 y172 ff8 fs6 fc1 sc0 ls125">A<span class="ls126 v15">\ue012</span><span class="ff4 lsb4 v2">c</span><span class="ff1 fs2 ls2f v16">1</span></div><div class="t m0 x49 h47 y173 ff4 fs6 fc1 sc0 lsb4">c<span class="ff1 fs2 lsd9 v19">2</span><span class="ff8 lsda v13">\ue013</span><span class="ff7 fs7 lsb6 vd">=</span><span class="ff8 ls2f v17">0</span></div><div class="t m0 x5b h17 y174 ff8 fs6 fc1 sc0 ls2f">@</div><div class="t m0 x41 h15 y175 ff1 fs6 fc1 sc0 ls2f">1</div><div class="t m0 x3b h27 y176 ff5 fs7 fc1 sc0 ls121">\ue000<span class="ff1 fs6 ls2f">1</span></div><div class="t m0 x41 h15 y177 ff1 fs6 fc1 sc0 ls2f">2</div><div class="t m0 x5c h17 y178 ff8 fs6 fc1 sc0 ls2f">1</div><div class="t m0 x5c h17 y179 ff8 fs6 fc1 sc0 ls2f">A</div><div class="t m0 xd h15 y17a ff1 fs6 fc1 sc0 ls2f ws8d">deve<span class="_b blank"> </span>p ossuir<span class="_6 blank"> </span>uma<span class="_b blank"> </span>solução.</div><div class="t m0 x25 h5 y17b ff1 fs2 fc1 sc0 ls2f ws3c">21</div></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf8" class="pf w0 h0" data-page-no="8"><div class="pc pc8 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x0 y17c w1 h48" alt="" src="https://files.passeidireto.com/f576e19b-d334-4686-aa50-430d24758f0a/bg8.png"><div class="c x0 y1 w2 h2"><div class="t m0 x6 h3 y19 ff1 fs0 fc0 sc0 ls2f ws4">Ex<span class="_0 blank"></span>erc<span class="_0 blank"></span>ício<span class="_0 blank"></span>s</div><div class="t m0 x7 h4 y17d ff1 fs1 fc1 sc0 ls2f ws3">(Cont.)</div><div class="t m0 xc h49 y17e ffb fs5 fc0 sc0 ls4f">I<span class="ff1 fs6 fc1 ls2f ws61 v1">Escalonando a matriz de co<span class="_4 blank"> </span>e\u2026cientes ampliada, obtemos que:</span></div><div class="t m0 x1f h17 y17f ff8 fs6 fc1 sc0 ls2f">b</div><div class="t m0 x5d h4a y180 ff4 fs6 fc1 sc0 ls127">A<span class="ff7 fs7 ls54">=</span><span class="ff8 ls2f v13">0</span></div><div class="t m0 x47 h17 y181 ff8 fs6 fc1 sc0 ls2f">@</div><div class="t m0 x4f h15 y182 ff1 fs6 fc1 sc0 ls2f ws64">1 3<span class="_11 blank"> </span>1</div><div class="t m0 x4f h27 y183 ff1 fs6 fc1 sc0 ls2f ws64">2 2 <span class="ff5 fs7 ls128">\ue000</span>1</div><div class="t m0 x4f h15 y184 ff1 fs6 fc1 sc0 ls2f ws64">3 1<span class="_11 blank"> </span>2</div><div class="t m0 x49 h17 y185 ff8 fs6 fc1 sc0 ls2f">1</div><div class="t m0 x49 h3a y186 ff8 fs6 fc1 sc0 lsb5">A<span class="ff5 fs7 ls129 v9">)</span><span class="ls2f v14">0</span></div><div class="t m0 x4a h17 y187 ff8 fs6 fc1 sc0 ls2f">@</div><div class="t m0 x4b h15 y188 ff1 fs6 fc1 sc0 ls2f ws8c">1 2<span class="_11 blank"> </span>1</div><div class="t m0 x4b h4b y189 ff1 fs6 fc1 sc0 ls2f ws8c">0 1<span class="_12 blank"> </span><span class="fs2 v2">3</span></div><div class="t m0 x5a h5 y18a ff1 fs2 fc1 sc0 ls2f">4</div><div class="t m0 x4b h27 y18b ff1 fs6 fc1 sc0 ls2f ws64">0 0 <span class="ff5 fs7 ls12a">\ue000</span>5</div><div class="t m0 x5e h17 y18c ff8 fs6 fc1 sc0 ls2f">1</div><div class="t m0 x5e h17 y18d ff8 fs6 fc1 sc0 ls2f">A</div><div class="t m0 xd h4c y18e ff1 fs6 fc1 sc0 ls2f ws8e">P<span class="_0 blank"></span>ortanto, <span class="ff4 ws31">posto<span class="_10 blank"> </span>A<span class="_13 blank"> </span><span class="ff9 fs7 ls12b"><</span><span class="ws2f">p osto<span class="_10 blank"> </span><span class="ff8 v16">b</span></span></span></div><div class="t m0 x15 h15 y18e ff4 fs6 fc1 sc0 ls2f ws2d">A<span class="ff1 ws61">.<span class="_10 blank"> </span>Logo, o sistema não p<span class="_4 blank"> </span>ossui solução.</span></div><div class="t m0 xc h4d y18f ffb fs5 fc0 sc0 ls4f">I<span class="ff1 fs6 fc1 ls2f ws8f v1">Assim, concluímos que <span class="ff2 ls12c">w<span class="ffa ls2f ws90">/<span class="_14 blank"></span><span class="ff5 fs7 ws91">2<span class="_b blank"> </span>L<span class="ff7 ls11e">[</span><span class="ff2 fs6 ws2d">u<span class="ff3 lsb9">,</span><span class="ls12d">v</span></span><span class="ff7 ls11e">]</span><span class="ff3 fs6">.</span></span></span></span></span></div><div class="t m0 x25 h5 y6 ff1 fs2 fc1 sc0 ls2f ws3c">22</div></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div>
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