<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/29c85f5a-8b02-49ce-9165-c9b1cf03d469/bg1.png"><div class="c x0 y1 w2 h2"><div class="t m0 x1 h3 y2 ff1 fs0 fc0 sc0 ls39 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 ls39 ws1">Aula 11:<span class="_2 blank"> </span>T<span class="_3 blank"></span>ransfo<span class="_0 blank"></span>rmações Lineares</div><div class="t m0 x3 h4 y4 ff1 fs1 fc1 sc0 ls39 ws2">Ma<span class="_0 blank"></span>rcos Y. Nakaguma</div><div class="t m0 x4 h4 y5 ff1 fs1 fc1 sc0 ls39 ws3">13/09/2017</div><div class="t m0 x5 h5 y6 ff1 fs2 fc1 sc0 ls39">1</div></div><div class="c x0 y7 w2 h2"><div class="t m0 x6 h3 y8 ff1 fs0 fc0 sc0 ls39 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 ls39 ws5">Dada uma matriz de co<span class="_4 blank"> </span>e\u2026cientes <span class="ff2 ls0">n<span class="ff3 fs3 ls1">\ue002</span><span class="ls2">m</span></span>:</div><div class="t m0 x8 h6 ya ff2 fs1 fc1 sc0 ls3">A<span class="ff4 fs3 ls4 v0">=</span><span class="ff5 ls39 v1">0</span></div><div class="t m0 x9 h7 yb ff5 fs1 fc1 sc0 ls39">B</div><div class="t m0 x9 h7 yc ff5 fs1 fc1 sc0 ls39">@</div><div class="t m0 xa h8 yd ff2 fs1 fc1 sc0 ls5">a<span class="ff1 fs2 ls39 ws6 v2">11<span class="_5 blank"> </span></span><span class="ff3 fs3 ls39 ws7 v0">\ue001 \ue001 \ue001<span class="_6 blank"> </span></span><span class="ls6 v0">a<span class="ff1 fs2 ls7 v2">1<span class="ff2 ls39">m</span></span></span></div><div class="t m0 xb h4 ye ff1 fs1 fc1 sc0 ls39">.</div><div class="t m0 xb h4 yf ff1 fs1 fc1 sc0 ls39">.</div><div class="t m0 xb h9 y10 ff1 fs1 fc1 sc0 ls8">.<span class="ls9 v3">.</span><span class="lsa v4">.</span><span class="lsb v5">.</span><span class="ls39 v6">.</span></div><div class="t m0 xc h4 yf ff1 fs1 fc1 sc0 ls39">.</div><div class="t m0 xc h4 y10 ff1 fs1 fc1 sc0 ls39">.</div><div class="t m0 xa ha y11 ff2 fs1 fc1 sc0 ls5">a<span class="fs2 lsc v2">n<span class="ff1 lsd">1<span class="ff3 fs3 ls39 ws7 v7">\ue001 \ue001 \ue001<span class="_6 blank"> </span></span></span></span><span class="v0">a<span class="fs2 ls39 ws8 v2">nm</span></span></div><div class="t m0 xd h7 y12 ff5 fs1 fc1 sc0 ls39">1</div><div class="t m0 xd h7 y13 ff5 fs1 fc1 sc0 ls39">C</div><div class="t m0 xd hb y14 ff5 fs1 fc1 sc0 lse">A<span class="ff4 fs3 ls4 v5">=</span><span class="lsf v8">\ue002</span><span class="ff6 ls39 ws3 v5">a</span><span class="ff1 fs2 ls10 v9">1</span><span class="ff7 ls39 ws9 v5">, ..., <span class="ff6 ws3">a<span class="ff2 fs2 ls11 v2">m</span><span class="ff5 va">\ue003</span></span></span></div><div class="t m0 x7 h4 y15 ff1 fs1 fc1 sc0 ls39 wsa">o<span class="_7 blank"> </span>sub conjunto<span class="_7 blank"> </span>de<span class="_7 blank"> </span><span class="ff8 ls12">R<span class="ff2 fs2 ls13 v4">n</span></span><span class="wsb">gerado<span class="_7 blank"> </span>p elas<span class="_7 blank"> </span>colunas<span class="_7 blank"> </span>de<span class="_7 blank"> </span><span class="ff2 ls14">A</span><span class="wsc">é denominado</span></span></div><div class="t m0 x7 h4 y16 ff1 fs1 fc2 sc0 ls39 wsd">espaço-coluna <span class="fc1 wse">de <span class="ff2 ws3">A</span>:</span></div><div class="t m0 xe hc y17 ff2 fs1 fc1 sc0 ls39 wsf">Col <span class="ff4 fs3 ls15 v0">(</span><span class="ls16 v0">A<span class="ff4 fs3 ls17 v0">)<span class="ls18 v0">=<span class="ff3 ls19">L</span><span class="ls1a">[</span></span></span><span class="ff6 ls39 ws3">a<span class="ff1 fs2 ls1b v2">1</span><span class="ff7 ws10">, ..., </span>a</span><span class="fs2 ls1c v2">m</span><span class="ff4 fs3 ls39">]</span></span></div><div class="t m0 x7 h4 y18 ff1 fs1 fc1 sc0 ls39 ws11">P<span class="_0 blank"></span>or sua vez, a conjunto das soluções do sistema homogêneo <span class="ff2 ws3">A<span class="ff6 ls1d">x<span class="ff4 fs3 ls1e">=</span><span class="ls1f">0</span></span></span>é</div><div class="t m0 x7 h4 y19 ff1 fs1 fc1 sc0 ls39 ws12">denominado <span class="fc2 ws13">espaço-nulo </span><span class="ws14">de <span class="ff2 ws3">A</span><span class="ls20">,</span><span class="ff2 ws15">Nul <span class="ff4 fs3 ls21 v0">(</span><span class="ls22">A<span class="ff4 fs3 ls23 v0">)</span></span><span class="ff7">.</span></span></span></div><div class="t m0 x7 h4 y1a ff1 fs1 fc1 sc0 ls39 ws16">Na aula passada, vimos que o <span class="fc2 ws2">teo<span class="_0 blank"></span>rema fundamental da álgebra linea<span class="_0 blank"></span>r</span></div><div class="t m0 x7 h4 y1b ff1 fs1 fc1 sc0 ls39 ws17">nos ga<span class="_0 blank"></span>rante que dada uma matriz <span class="ff2 ls14">A</span><span class="ws2">de tamanho <span class="ff2 ls24">n<span class="ff3 fs3 ls1">\ue002</span><span class="ls25">m<span class="ff7 ls26">,</span></span></span>então:</span></div><div class="t m0 xf hd y1c ff1 fs1 fc1 sc0 ls39 ws18">dim <span class="ff2 ws19">Col <span class="ff4 fs3 ls15 v0">(</span><span class="ls16 v0">A<span class="ff4 fs3 ls27 v0">)<span class="ls28 v0">+</span></span></span></span><span class="ws1a v0">dim <span class="ff2 ws1b">Nul <span class="ff4 fs3 ls29 v0">(</span><span class="ls2a">A<span class="ff4 fs3 ls2b v0">)<span class="ls2c v0">=</span></span><span class="ls2">m</span></span><span class="ff7">,</span></span></span></div><div class="t m0 x7 h4 y1d ff1 fs1 fc1 sc0 ls39 ws1c">onde <span class="ff2 ls2d">m</span><span class="ws1d">é o número de colunas da matriz <span class="ff2 ws3">A</span>.</span></div><div class="t m0 x5 h5 y1e ff1 fs2 fc1 sc0 ls39">2</div></div><div class="c x0 y1f w2 h2"><div class="t m0 x6 h3 y20 ff1 fs0 fc0 sc0 ls39 ws1e">Ex<span class="_0 blank"></span>erc<span class="_0 blank"></span>ício 1<span class="_0 blank"></span>:<span class="_1 blank"> </span>27<span class="_0 blank"></span>.12(<span class="_0 blank"></span>d)</div><div class="t m0 x7 h4 y21 ff1 fs1 fc2 sc0 ls39 ws1f">Exercício: <span class="fc1 ws1d">Calcule uma base do espaço-nulo da seguinte matriz:</span></div><div class="t m0 xb he y22 ff2 fs1 fc1 sc0 ls2e">A<span class="ff4 fs3 ls4 v0">=</span><span class="ff5 ls39 vb">0</span></div><div class="t m0 x10 h7 y23 ff5 fs1 fc1 sc0 ls39">@</div><div class="t m0 x11 hd y24 ff1 fs1 fc1 sc0 ls39 ws20">4<span class="_8 blank"> </span>1 <span class="ff3 fs3 ls2f v0">\ue000</span><span class="ws21 v0">5 1</span></div><div class="t m0 x11 h4 y25 ff1 fs1 fc1 sc0 ls39 ws22">8<span class="_8 blank"> </span>5 <span class="ff3 fs3 ls30 v0">\ue000</span><span class="ws23 v0">10 8</span></div><div class="t m0 x12 h4 y26 ff3 fs3 fc1 sc0 ls31">\ue000<span class="ff1 fs1 ls39 ws23">4 2<span class="_9 blank"> </span>7<span class="_9 blank"> </span>5</span></div><div class="t m0 x13 h7 y27 ff5 fs1 fc1 sc0 ls39">1</div><div class="t m0 x13 h7 y28 ff5 fs1 fc1 sc0 ls39">A</div><div class="t m0 x14 hf y29 ff9 fs4 fc0 sc0 ls32">I<span class="ff1 fs5 fc1 ls39 ws24 v2">O espaço-nulo de <span class="ff2 ls33">A</span><span class="ws25">é dado p<span class="_4 blank"> </span>elo conjunto de soluções do sistema de</span></span></div><div class="t m0 x15 h10 y2a ff1 fs5 fc1 sc0 ls39 ws26">equações homogêneo <span class="ff2 ws27">A<span class="ff6 ls34">x<span class="ff4 fs6 ls35">=</span><span class="ls39">0<span class="ff7">.</span></span></span></span></div><div class="t m0 x14 h11 y2b ff9 fs4 fc0 sc0 ls32">I<span class="ff1 fs5 fc1 ls39 ws28 v2">Note que a matriz <span class="ff2 ls33">A</span><span class="ws29">em sua forma escalonada por linhas é dada por:</span></span></div><div class="t m0 x16 h12 y2c ff2 fs5 fc1 sc0 ls39 ws27">A<span class="fs2 ls36 v2">r</span><span class="ff4 fs6 ls37">=</span><span class="ff5 vc">0</span></div><div class="t m0 x17 h13 y2d ff5 fs5 fc1 sc0 ls39">@</div><div class="t m0 x18 h10 y2e ff1 fs5 fc1 sc0 ls39 ws2a">4 1 <span class="ff3 fs6 ls38">\ue000</span>5 1</div><div class="t m0 x18 h10 y2f ff1 fs5 fc1 sc0 ls39 ws2b">0<span class="_a blank"> </span>3 0 6</div><div class="t m0 x18 h14 y30 ff1 fs5 fc1 sc0 ls39 ws2b">0<span class="_a blank"> </span>0 2 0<span class="_b blank"> </span><span class="ff5 vd">1</span></div><div class="t m0 x19 h13 y31 ff5 fs5 fc1 sc0 ls39">A</div><div class="t m0 x5 h5 y32 ff1 fs2 fc1 sc0 ls39">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/29c85f5a-8b02-49ce-9165-c9b1cf03d469/bg2.png"><div class="c x0 y1 w2 h2"><div class="t m0 x6 h3 y33 ff1 fs0 fc0 sc0 ls39 ws1e">Ex<span class="_0 blank"></span>erc<span class="_0 blank"></span>ício 1<span class="_0 blank"></span>:<span class="_1 blank"> </span>27<span class="_0 blank"></span>.12(<span class="_0 blank"></span>d)</div><div class="t m0 x7 h4 y34 ff1 fs1 fc1 sc0 ls39 ws3">(Cont.)</div><div class="t m0 x14 hf y35 ff9 fs4 fc0 sc0 ls32">I<span class="ff1 fs5 fc1 ls39 ws25 v2">Assim, o sistema <span class="ff2 ws27">A<span class="ff6 ls3a">x<span class="ff4 fs6 ls35">=</span><span class="ls3b">0</span></span><span class="ff1 ws2d">p o de<span class="_c blank"> </span>ser<span class="_c blank"> </span>exp<span class="_0 blank"></span>resso<span class="_c blank"> </span>como:</span></span></span></div><div class="t m0 x1a h13 y36 ff5 fs5 fc1 sc0 ls39">8</div><div class="t m0 x1a h13 y37 ff5 fs5 fc1 sc0 ls39"><</div><div class="t m0 x1a h13 y38 ff5 fs5 fc1 sc0 ls39">:</div><div class="t m0 x1b h10 y39 ff1 fs5 fc1 sc0 ls39 ws27">4<span class="ff2 ls3c">x</span><span class="fs2 ls3d v2">1</span><span class="ff4 fs6 ls3e">+</span><span class="ff2 ls3f">x</span><span class="fs2 ls40 v2">2</span><span class="ff3 fs6 ls41">\ue000</span>5<span class="ff2">x</span><span class="fs2 ls42 v2">3</span><span class="ff4 fs6 ls38">+</span><span class="ff2 ls3f">x</span><span class="fs2 ls40 v2">4</span><span class="ff4 fs6 ls35">=</span>0</div><div class="t m0 xa h10 y3a ff1 fs5 fc1 sc0 ls39 ws27">3<span class="ff2 ls3f">x</span><span class="fs2 ls43 v2">2</span><span class="ff4 fs6 ls38">+</span>6<span class="ff2 ls3f">x</span><span class="fs2 ls40 v2">4</span><span class="ff4 fs6 ls35">=</span>0</div><div class="t m0 x1c h10 y3b ff1 fs5 fc1 sc0 ls39 ws27">2<span class="ff2">x</span><span class="fs2 ls43 v2">3</span><span class="ff4 fs6 ls35">=</span>0</div><div class="t m0 x1d h15 y3c ff3 fs6 fc1 sc0 ls44">)<span class="ff5 fs5 ls39 ve">8</span></div><div class="t m0 x1e h13 y3d ff5 fs5 fc1 sc0 ls39"><</div><div class="t m0 x1e h13 y3e ff5 fs5 fc1 sc0 ls39">:</div><div class="t m0 x1f h16 y3f ff2 fs5 fc1 sc0 ls39 ws27">x<span class="ff1 fs2 ls45 v2">1</span><span class="ff4 fs6 ls46">=</span><span class="ff1 fs2 v4">1</span></div><div class="t m0 x20 h17 y40 ff1 fs2 fc1 sc0 ls47">4<span class="ff2 fs5 ls39 ws27 vf">x</span><span class="ls39 v7">4</span></div><div class="t m0 x1f h10 y41 ff2 fs5 fc1 sc0 ls39 ws27">x<span class="ff1 fs2 ls48 v2">2</span><span class="ff4 fs6 ls37">=<span class="ff3 ls38">\ue000</span></span><span class="ff1">2</span><span class="ls3f">x</span><span class="ff1 fs2 v2">4</span></div><div class="t m0 x1f h10 y42 ff2 fs5 fc1 sc0 ls39 ws27">x<span class="ff1 fs2 ls49 v2">3</span><span class="ff4 fs6 ls35">=</span><span class="ff1">0</span></div><div class="t m0 x15 h10 y43 ff1 fs5 fc1 sc0 ls39 ws2e">onde <span class="ff2 ws27">x</span><span class="fs2 ls4a v2">4</span><span class="ws2f">é a <span class="fc2 ws30 v0">va<span class="_0 blank"></span>riável livre <span class="fc1 ws25">do sistema.</span></span></span></div><div class="t m0 x14 h18 y44 ff9 fs4 fc0 sc0 ls32">I<span class="ff1 fs5 fc1 ls39 ws25 v2">P<span class="_0 blank"></span>ortanto, qualquer solução de <span class="ff2 ws27">A<span class="ff6 ls4b">x<span class="ff4 fs6 ls35">=</span><span class="ls4c">0</span></span><span class="ff1 ws2d">po de<span class="_c blank"> </span>ser<span class="_c blank"> </span>escrita<span class="_c blank"> </span>como:</span></span></span></div><div class="t m0 x21 h19 y45 ff6 fs5 fc1 sc0 ls4d">x<span class="ff4 fs6 ls4e">=</span><span class="ff5 ls39 v10">0</span></div><div class="t m0 xb h13 y46 ff5 fs5 fc1 sc0 ls39">B</div><div class="t m0 xb h13 y47 ff5 fs5 fc1 sc0 ls39">B</div><div class="t m0 xb h13 y48 ff5 fs5 fc1 sc0 ls39">@</div><div class="t m0 x22 h1a y49 ff2 fs5 fc1 sc0 ls39 ws27">x<span class="ff1 fs2 v2">1</span></div><div class="t m0 x22 h1a y4a ff2 fs5 fc1 sc0 ls39 ws27">x<span class="ff1 fs2 v2">2</span></div><div class="t m0 x22 h1a y4b ff2 fs5 fc1 sc0 ls39 ws27">x<span class="ff1 fs2 v2">3</span></div><div class="t m0 x22 h1a y4c ff2 fs5 fc1 sc0 ls39 ws27">x<span class="ff1 fs2 v2">4</span></div><div class="t m0 x1c h13 y4d ff5 fs5 fc1 sc0 ls39">1</div><div class="t m0 x1c h13 y4e ff5 fs5 fc1 sc0 ls39">C</div><div class="t m0 x1c h13 y4f ff5 fs5 fc1 sc0 ls39">C</div><div class="t m0 x1c h1b y50 ff5 fs5 fc1 sc0 ls4f">A<span class="ff4 fs6 ls50 v4">=</span><span class="ls39 v11">0</span></div><div class="t m0 xc h13 y51 ff5 fs5 fc1 sc0 ls39">B</div><div class="t m0 xc h13 y52 ff5 fs5 fc1 sc0 ls39">B</div><div class="t m0 xc h13 y53 ff5 fs5 fc1 sc0 ls39">@</div><div class="t m0 x23 h5 y54 ff1 fs2 fc1 sc0 ls39">1</div><div class="t m0 x23 h1c y55 ff1 fs2 fc1 sc0 ls47">4<span class="ff2 fs5 ls39 ws27 vf">x</span><span class="ls39 v7">4</span></div><div class="t m0 x24 h10 y56 ff3 fs6 fc1 sc0 ls38">\ue000<span class="ff1 fs5 ls39 ws27">2<span class="ff2">x</span><span class="fs2 v2">4</span></span></div><div class="t m0 x25 h10 y57 ff1 fs5 fc1 sc0 ls39">0</div><div class="t m0 x26 h1a y58 ff2 fs5 fc1 sc0 ls39 ws27">x<span class="ff1 fs2 v2">4</span></div><div class="t m0 x27 h13 y59 ff5 fs5 fc1 sc0 ls39">1</div><div class="t m0 x27 h13 y5a ff5 fs5 fc1 sc0 ls39">C</div><div class="t m0 x27 h13 y5b ff5 fs5 fc1 sc0 ls39">C</div><div class="t m0 x27 h1d y5c ff5 fs5 fc1 sc0 ls51">A<span class="ff4 fs6 ls52 v4">=</span><span class="ff2 ls39 ws27 v4">x</span><span class="ff1 fs2 ls53 v12">4</span><span class="ls39 v11">0</span></div><div class="t m0 x28 h13 y5d ff5 fs5 fc1 sc0 ls39">B</div><div class="t m0 x28 h13 y5e ff5 fs5 fc1 sc0 ls39">B</div><div class="t m0 x28 h13 y5f ff5 fs5 fc1 sc0 ls39">@</div><div class="t m0 x29 h5 y60 ff1 fs2 fc1 sc0 ls39">1</div><div class="t m0 x29 h5 y55 ff1 fs2 fc1 sc0 ls39">4</div><div class="t m0 x2a h10 y61 ff3 fs6 fc1 sc0 ls54">\ue000<span class="ff1 fs5 ls39">2</span></div><div class="t m0 x29 h10 y62 ff1 fs5 fc1 sc0 ls39">0</div><div class="t m0 x29 h10 y63 ff1 fs5 fc1 sc0 ls39">1</div><div class="t m0 x2b h13 y64 ff5 fs5 fc1 sc0 ls39">1</div><div class="t m0 x2b h13 y65 ff5 fs5 fc1 sc0 ls39">C</div><div class="t m0 x2b h13 y66 ff5 fs5 fc1 sc0 ls39">C</div><div class="t m0 x2b h13 y67 ff5 fs5 fc1 sc0 ls39">A</div><div class="t m0 x15 h1e y68 ff1 fs5 fc1 sc0 ls39 ws25">e o veto<span class="_0 blank"></span>r<span class="_7 blank"> </span><span class="ff4 fs6 ls55">(</span><span class="fs2 v4">1</span></div><div class="t m0 x2c h1f y69 ff1 fs2 fc1 sc0 ls47">4<span class="ff7 fs5 ls56 vf">,<span class="ff3 fs6 ls38">\ue000<span class="ff1 fs5 ls57">2<span class="ff7 ls58">,</span><span class="ls39 ws27">0<span class="ff7 ls59">,</span><span class="ls5a">1</span></span></span><span class="ff4 ls5b">)<span class="ff1 fs5 ls39 ws30">constitui uma <span class="fc2 ws31 v0">base <span class="fc1 ws32">de <span class="ff2 ws33">Nul </span></span></span></span><span class="ls5c v0">(<span class="ff2 fs5 ls5d v0">A</span><span class="ls5e">)</span></span></span></span></span><span class="fs5 ls39 ws34 vf">, de fo<span class="_0 blank"></span>rma que</span></div><div class="t m0 x15 h10 y6a ff1 fs5 fc1 sc0 ls39 ws35">dim <span class="ff2 ws33">Nul <span class="ff4 fs6 ls5e v0">(</span><span class="ls5f">A<span class="ff4 fs6 ls60 v0">)<span class="ls35 v0">=</span></span></span></span><span class="ws27">1<span class="ff7">.</span></span></div><div class="t m0 x5 h5 y6b ff1 fs2 fc1 sc0 ls39">4</div></div><div class="c x0 y7 w2 h2"><div class="t m0 x6 h3 y8 ff1 fs0 fc0 sc0 ls39 ws1e">Ex<span class="_0 blank"></span>erc<span class="_0 blank"></span>ício 1<span class="_0 blank"></span>:<span class="_1 blank"> </span>27<span class="_0 blank"></span>.12(<span class="_0 blank"></span>d)</div><div class="t m0 x7 h4 y6c ff1 fs1 fc1 sc0 ls39 ws3">(Cont.)</div><div class="t m0 x14 hf y6d ff9 fs4 fc0 sc0 ls32">I<span class="ff1 fs5 fc1 ls39 ws25 v2">Assim como anterio<span class="_0 blank"></span>rmente, note que:</span></div><div class="t m0 x2d h20 y6e ff1 fs5 fc1 sc0 ls39 ws35">dim <span class="ff2 ws36">Nul <span class="ff4 fs6 ls61 v0">(</span><span class="ls62">A<span class="ff4 fs6 ls63 v0">)<span class="ls64 v0">=</span></span><span class="ls65">n<span class="fs2 ls66 v4">o</span></span></span></span><span class="ws29">de va<span class="_0 blank"></span>riáveis livres</span></div><div class="t m0 x2e h21 y6f ff4 fs6 fc1 sc0 ls64">=<span class="ff2 fs5 ls65">n<span class="fs2 ls66 v4">o</span><span class="ff1 ls39 ws37">de va<span class="_0 blank"></span>riáveis<span class="_d blank"> </span><span class="ff3 fs6 ls67">\ue000</span><span class="ws38">p osto<span class="_c blank"> </span><span class="ff2">A</span></span></span></span></div><div class="t m0 x2e h20 y70 ff4 fs6 fc1 sc0 ls64">=<span class="ff2 fs5 ls65">n<span class="fs2 ls66 v4">o</span><span class="ff1 ls39 ws37">de va<span class="_0 blank"></span>riáveis<span class="_d blank"> </span><span class="ff3 fs6 ls67">\ue000</span><span class="ff2 ws39">Col <span class="ff4 fs6 ls5e v0">(</span><span class="ls68">A</span><span class="ff4 fs6 v0">)</span></span></span></span></div><div class="t m0 x2e h10 y71 ff4 fs6 fc1 sc0 ls64">=<span class="ff1 fs5 ls69">4</span><span class="ff3 ls6a">\ue000<span class="ff1 fs5 ls34">3</span></span><span class="ls35">=<span class="ff1 fs5 ls39">1</span></span></div><div class="t m0 x5 h5 y72 ff1 fs2 fc1 sc0 ls39">5</div></div><div class="c x0 y1f w2 h2"><div class="t m0 x2c h22 y73 ff1 fs7 fc1 sc0 ls39 ws3a">T<span class="_e blank"></span>r<span class="_0 blank"></span>a<span class="_0 blank"></span>n<span class="_0 blank"></span>s<span class="_0 blank"></span>fo<span class="_3 blank"></span>r<span class="_0 blank"></span>m<span class="_3 blank"></span>a<span class="_0 blank"></span>ç<span class="_0 blank"></span>õ<span class="_0 blank"></span>e<span class="_0 blank"></span>s L<span class="_0 blank"></span>i<span class="_0 blank"></span>n<span class="_0 blank"></span>e<span class="_0 blank"></span>a<span class="_3 blank"></span>r<span class="_0 blank"></span>e<span class="_0 blank"></span>s</div><div class="t m0 x5 h5 y74 ff1 fs2 fc1 sc0 ls39">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/29c85f5a-8b02-49ce-9165-c9b1cf03d469/bg3.png"><div class="c x0 y1 w2 h2"><div class="t m0 x6 h3 y33 ff1 fs0 fc0 sc0 ls39 ws0">Re<span class="_0 blank"></span>vis<span class="_0 blank"></span>ão:<span class="_f blank"> </span>Á<span class="_0 blank"></span>lgeb<span class="_3 blank"></span>ra Li<span class="_0 blank"></span>nea<span class="_3 blank"></span>r</div><div class="t m0 x7 h4 y75 ff1 fs1 fc1 sc0 ls39 wsc">A<span class="_0 blank"></span>té o momento neste curso, estudamos como analisar <span class="fc2 ws17">sistemas de</span></div><div class="t m0 x7 h4 y76 ff1 fs1 fc2 sc0 ls39 ws3b">equações linea<span class="_0 blank"></span>res <span class="fc1 ws3c v0">com<span class="_7 blank"> </span>base<span class="_7 blank"> </span>em<span class="_7 blank"> </span>duas<span class="_7 blank"> </span>ab ordagens:</span></div><div class="t m0 x2f h10 y77 ff2 fs5 fc0 sc0 ls6b">i<span class="ff7 ls6c">.<span class="ff1 fc2 ls39 ws2d">Ab o<span class="_0 blank"></span>rdagem<span class="_c blank"> </span>p or<span class="_c blank"> </span>Linhas:</span></span></div><div class="t m0 xa h13 y78 ff5 fs5 fc1 sc0 ls39">8</div><div class="t m0 xa h13 y79 ff5 fs5 fc1 sc0 ls39">></div><div class="t m0 xa h13 y7a ff5 fs5 fc1 sc0 ls39">></div><div class="t m0 xa h13 y7b ff5 fs5 fc1 sc0 ls39">></div><div class="t m0 xa h13 y7c ff5 fs5 fc1 sc0 ls39"><</div><div class="t m0 xa h13 y7d ff5 fs5 fc1 sc0 ls39">></div><div class="t m0 xa h13 y7e ff5 fs5 fc1 sc0 ls39">></div><div class="t m0 xa h13 y7f ff5 fs5 fc1 sc0 ls39">></div><div class="t m0 xa h13 y80 ff5 fs5 fc1 sc0 ls39">:</div><div class="t m0 x30 h23 y81 ff2 fs5 fc1 sc0 ls6d">a<span class="ff1 fs2 ls39 ws6 v2">11<span class="_10 blank"> </span></span><span class="ls39 ws27">x<span class="ff1 fs2 ls6e v2">1</span><span class="ff4 fs6 ls6a">+</span><span class="ls6f">a</span><span class="ff1 fs2 ws6 v2">12<span class="_10 blank"> </span></span>x<span class="ff1 fs2 ls70 v2">2</span><span class="ff4 fs6 ls71">+</span><span class="ff7 ws3d">... <span class="ff4 fs6 ls72">+</span></span><span class="ls73">a<span class="ff1 fs2 ls74 v13">1<span class="ff2 ls75">k</span></span></span>x<span class="fs2 ls76 v13">k</span><span class="ff4 fs6 ls35">=</span>b<span class="ff1 fs2 v2">1</span></span></div><div class="t m0 x30 h23 y82 ff2 fs5 fc1 sc0 ls6d">a<span class="ff1 fs2 ls39 ws6 v2">21<span class="_10 blank"> </span></span><span class="ls39 ws27">x<span class="ff1 fs2 ls6e v2">1</span><span class="ff4 fs6 ls6a">+</span><span class="ls6f">a</span><span class="ff1 fs2 ws6 v2">22<span class="_10 blank"> </span></span>x<span class="ff1 fs2 ls70 v2">2</span><span class="ff4 fs6 ls71">+</span><span class="ff7 ws3d">... <span class="ff4 fs6 ls72">+</span></span><span class="ls73">a<span class="ff1 fs2 ls74 v13">2<span class="ff2 ls75">k</span></span></span>x<span class="fs2 ls76 v13">k</span><span class="ff4 fs6 ls35">=</span>b<span class="ff1 fs2 v2">2</span></span></div><div class="t m0 x31 h10 y83 ff1 fs5 fc1 sc0 ls39">.</div><div class="t m0 x31 h10 y84 ff1 fs5 fc1 sc0 ls39">.</div><div class="t m0 x31 h10 y85 ff1 fs5 fc1 sc0 ls39">.</div><div class="t m0 x30 h23 y86 ff2 fs5 fc1 sc0 ls77">a<span class="fs2 ls78 v2">n<span class="ff1 ls79">1</span></span><span class="ls39 ws27">x<span class="ff1 fs2 ls6e v2">1</span><span class="ff4 fs6 ls7a">+</span><span class="ls7b">a<span class="fs2 lsc v2">n<span class="ff1 ls10">2</span></span><span class="ls3f">x<span class="ff1 fs2 ls6e v2">2</span><span class="ff4 fs6 ls72">+</span></span></span><span class="ff7 ws3d">... <span class="ff4 fs6 ls6a">+</span></span><span class="ls6f">a</span><span class="fs2 ws3e v13">nk<span class="_11 blank"> </span></span>x<span class="fs2 ls76 v13">k</span><span class="ff4 fs6 ls35">=</span>b<span class="fs2 v2">n</span></span></div><div class="t m0 x32 h10 y87 ff2 fs5 fc0 sc0 ls39 ws3f">ii <span class="ff7 ls7c">.</span><span class="ff1 fc2 ws2d">Ab o<span class="_0 blank"></span>rdagem<span class="_c blank"> </span>p or<span class="_c blank"> </span>Colunas:</span></div><div class="t m0 x1a h1a y88 ff2 fs5 fc1 sc0 ls39 ws27">x<span class="ff1 fs2 v2">1</span></div><div class="t m0 x33 h13 y89 ff5 fs5 fc1 sc0 ls39">2</div><div class="t m0 x33 h13 y8a ff5 fs5 fc1 sc0 ls39">6</div><div class="t m0 x33 h13 y8b ff5 fs5 fc1 sc0 ls39">6</div><div class="t m0 x33 h13 y8c ff5 fs5 fc1 sc0 ls39">6</div><div class="t m0 x33 h13 y8d ff5 fs5 fc1 sc0 ls39">4</div><div class="t m0 x2 h1a y8e ff2 fs5 fc1 sc0 ls6d">a<span class="ff1 fs2 ls39 ws3e v2">11</span></div><div class="t m0 x2 h1a y8f ff2 fs5 fc1 sc0 ls6d">a<span class="ff1 fs2 ls39 ws3e v2">21</span></div><div class="t m0 x9 h10 y90 ff1 fs5 fc1 sc0 ls39">.</div><div class="t m0 x9 h10 y91 ff1 fs5 fc1 sc0 ls39">.</div><div class="t m0 x9 h10 y92 ff1 fs5 fc1 sc0 ls39">.</div><div class="t m0 x2 h1a y93 ff2 fs5 fc1 sc0 ls73">a<span class="fs2 ls7d v2">n<span class="ff1 ls39">1</span></span></div><div class="t m0 xa h13 y94 ff5 fs5 fc1 sc0 ls39">3</div><div class="t m0 xa h13 y95 ff5 fs5 fc1 sc0 ls39">7</div><div class="t m0 xa h13 y96 ff5 fs5 fc1 sc0 ls39">7</div><div class="t m0 xa h13 y97 ff5 fs5 fc1 sc0 ls39">7</div><div class="t m0 xa h24 y98 ff5 fs5 fc1 sc0 ls7e">5<span class="ff4 fs6 ls72 v3">+</span><span class="ff2 ls3f v3">x</span><span class="ff1 fs2 ls39 v14">2</span></div><div class="t m0 x4 h13 y99 ff5 fs5 fc1 sc0 ls39">2</div><div class="t m0 x4 h13 y9a ff5 fs5 fc1 sc0 ls39">6</div><div class="t m0 x4 h13 y9b ff5 fs5 fc1 sc0 ls39">6</div><div class="t m0 x4 h13 y9c ff5 fs5 fc1 sc0 ls39">6</div><div class="t m0 x4 h13 y9d ff5 fs5 fc1 sc0 ls39">4</div><div class="t m0 x12 h1a y9e ff2 fs5 fc1 sc0 ls7b">a<span class="ff1 fs2 ls39 ws6 v2">12</span></div><div class="t m0 x12 h1a y9f ff2 fs5 fc1 sc0 ls7b">a<span class="ff1 fs2 ls39 ws6 v2">22</span></div><div class="t m0 x11 h10 ya0 ff1 fs5 fc1 sc0 ls39">.</div><div class="t m0 x11 h10 ya1 ff1 fs5 fc1 sc0 ls39">.</div><div class="t m0 x11 h10 ya2 ff1 fs5 fc1 sc0 ls39">.</div><div class="t m0 x34 h1a ya3 ff2 fs5 fc1 sc0 ls6f">a<span class="fs2 ls7d v2">n<span class="ff1 ls39">2</span></span></div><div class="t m0 x35 h13 ya4 ff5 fs5 fc1 sc0 ls39">3</div><div class="t m0 x35 h13 ya5 ff5 fs5 fc1 sc0 ls39">7</div><div class="t m0 x35 h13 ya6 ff5 fs5 fc1 sc0 ls39">7</div><div class="t m0 x35 h13 ya7 ff5 fs5 fc1 sc0 ls39">7</div><div class="t m0 x35 h24 ya8 ff5 fs5 fc1 sc0 ls7e">5<span class="ff4 fs6 ls7f v3">+<span class="ff3 ls39 ws40">\ue001 \ue001 \ue001<span class="_12 blank"> </span><span class="ff4 ls6a">+</span><span class="ff2 fs5 ws27">x<span class="fs2 v13">k</span></span></span></span></div><div class="t m0 x36 h13 ya4 ff5 fs5 fc1 sc0 ls39">2</div><div class="t m0 x36 h13 ya5 ff5 fs5 fc1 sc0 ls39">6</div><div class="t m0 x36 h13 ya6 ff5 fs5 fc1 sc0 ls39">6</div><div class="t m0 x36 h13 ya7 ff5 fs5 fc1 sc0 ls39">6</div><div class="t m0 x36 h13 ya8 ff5 fs5 fc1 sc0 ls39">4</div><div class="t m0 x1e h1a ya9 ff2 fs5 fc1 sc0 ls7b">a<span class="ff1 fs2 ls80 v13">1<span class="ff2 ls39">k</span></span></div><div class="t m0 x1e h1a yaa ff2 fs5 fc1 sc0 ls7b">a<span class="ff1 fs2 ls80 v13">2<span class="ff2 ls39">k</span></span></div><div class="t m0 x37 h10 yab ff1 fs5 fc1 sc0 ls39">.</div><div class="t m0 x37 h10 yac ff1 fs5 fc1 sc0 ls39">.</div><div class="t m0 x37 h10 yad ff1 fs5 fc1 sc0 ls39">.</div><div class="t m0 x1e h1a yae ff2 fs5 fc1 sc0 ls6d">a<span class="fs2 ls39 ws8 v13">nk</span></div><div class="t m0 x38 h13 yaf ff5 fs5 fc1 sc0 ls39">3</div><div class="t m0 x38 h13 yb0 ff5 fs5 fc1 sc0 ls39">7</div><div class="t m0 x38 h13 yb1 ff5 fs5 fc1 sc0 ls39">7</div><div class="t m0 x38 h13 yb2 ff5 fs5 fc1 sc0 ls39">7</div><div class="t m0 x38 h25 yb3 ff5 fs5 fc1 sc0 ls81">5<span class="ff4 fs6 ls37 v3">=</span><span class="ls39 v15">2</span></div><div class="t m0 x39 h13 yb4 ff5 fs5 fc1 sc0 ls39">6</div><div class="t m0 x39 h13 yb5 ff5 fs5 fc1 sc0 ls39">6</div><div class="t m0 x39 h13 yb6 ff5 fs5 fc1 sc0 ls39">6</div><div class="t m0 x39 h13 yb7 ff5 fs5 fc1 sc0 ls39">4</div><div class="t m0 x3a h1a yb8 ff2 fs5 fc1 sc0 ls39 ws27">b<span class="ff1 fs2 v2">1</span></div><div class="t m0 x3a h1a yb9 ff2 fs5 fc1 sc0 ls39 ws27">b<span class="ff1 fs2 v2">2</span></div><div class="t m0 x3b h10 yba ff1 fs5 fc1 sc0 ls39">.</div><div class="t m0 x3b h10 ybb ff1 fs5 fc1 sc0 ls39">.</div><div class="t m0 x3b h10 ybc ff1 fs5 fc1 sc0 ls39">.</div><div class="t m0 x3a h1a ybd ff2 fs5 fc1 sc0 ls39 ws27">b<span class="fs2 v2">n</span></div><div class="t m0 x3c h13 ybe ff5 fs5 fc1 sc0 ls39">3</div><div class="t m0 x3c h13 ybf ff5 fs5 fc1 sc0 ls39">7</div><div class="t m0 x3c h13 yc0 ff5 fs5 fc1 sc0 ls39">7</div><div class="t m0 x3c h13 yc1 ff5 fs5 fc1 sc0 ls39">7</div><div class="t m0 x3c h13 yc2 ff5 fs5 fc1 sc0 ls39">5</div><div class="t m0 x5 h5 yc3 ff1 fs2 fc1 sc0 ls39">7</div></div><div class="c x0 y7 w2 h2"><div class="t m0 x6 h3 y8 ff1 fs0 fc0 sc0 ls39 ws41">Re<span class="_0 blank"></span>vis<span class="_0 blank"></span>ão:<span class="_f blank"> </span>A<span class="_0 blank"></span>b o<span class="_0 blank"></span>rda<span class="_0 blank"></span>gem<span class="_13 blank"> </span>po<span class="_0 blank"></span>r<span class="_2 blank"> </span>L<span class="_0 blank"></span>inh<span class="_0 blank"></span>as</div><div class="t m0 x7 h4 yc4 ff1 fs1 fc1 sc0 ls39 ws1d">Dado um sistema de <span class="ff2 fc2 ls82">n</span><span class="ws42">equações e <span class="ff2 fc2 ls83">k</span></span>va<span class="_0 blank"></span>riáveis, mostramos que:</div><div class="t m0 x2f h26 yc5 ff2 fs5 fc0 sc0 ls6b">i<span class="ff7 ls6c">.<span class="ff1 fc1 ls39 ws2d">Se<span class="_c blank"> </span>p osto<span class="_c blank"> </span><span class="ff2 ls84">A<span class="ff4 fs6 ls85">=</span></span>posto<span class="_13 blank"> </span><span class="ff5 v12">b</span></span></span></div><div class="t m0 x22 h10 yc5 ff2 fs5 fc1 sc0 ls39 ws27">A<span class="ff1 ws25">, então o sistema <span class="fc2 ws37">admite solução</span>.</span></div><div class="t m0 x3d h27 yc6 ff2 fs8 fc0 sc0 ls86">a<span class="ffa ls87">.<span class="ff1 fc1 ls39 ws43">Se<span class="_c blank"> </span>p o s t o<span class="_c blank"> </span><span class="ff2 ls88">A<span class="ff4 fs9 ls89">=</span></span>p<span class="_4 blank"> </span>os t o<span class="_14 blank"> </span><span class="ff5 v12">b</span></span></span></div><div class="t m0 x1c h28 yc6 ff2 fs8 fc1 sc0 ls8a">A<span class="ff4 fs9 ls89">=</span><span class="ls8b">k<span class="ff1 ls39 ws44">,<span class="_c blank"> </span>então<span class="_c blank"> </span>ex i s te<span class="_c blank"> </span>um<span class="_4 blank"> </span>a<span class="_c blank"> </span><span class="fc2 ws45">únic a<span class="_c blank"> </span><span class="fc1">so lu ç ã o .</span></span></span></span></div><div class="t m0 x3d h29 yc7 ff2 fs8 fc0 sc0 ls8c">b<span class="ffa ls8d">.<span class="ff1 fc1 ls39 ws43">Se<span class="_c blank"> </span>p o s t o<span class="_c blank"> </span><span class="ff2 ls88">A<span class="ff4 fs9 ls89">=</span></span>p<span class="_4 blank"> </span>os t o<span class="_14 blank"> </span><span class="ff5 v12">b</span></span></span></div><div class="t m0 x1c h28 yc7 ff2 fs8 fc1 sc0 ls8a">A<span class="ffb fs9 ls89"><</span><span class="ls8b">k<span class="ff1 ls39 ws44">,<span class="_c blank"> </span>então<span class="_c blank"> </span>ex i s te m<span class="_c blank"> </span><span class="fc2 ws45">in \u2026 n i ta s<span class="_c blank"> </span></span><span class="ws43">so lu ç õ e s .</span></span></span></div><div class="t m0 x32 h26 yc8 ff2 fs5 fc0 sc0 ls39 ws46">ii <span class="ff7 ls8e">.</span><span class="ff1 fc1 ws2d">Se<span class="_c blank"> </span>p osto<span class="_c blank"> </span><span class="ff2 ls84">A<span class="ffb fs6 ls85"><</span></span>posto<span class="_14 blank"> </span><span class="ff5 v12">b</span></span></div><div class="t m0 x22 h10 yc8 ff2 fs5 fc1 sc0 ls39 ws27">A<span class="ff1 ws25">, então o sistema <span class="fc2 ws34">não admite solução</span>.</span></div><div class="t m0 x5 h5 yc9 ff1 fs2 fc1 sc0 ls39">8</div></div><div class="c x0 y1f w2 h2"><div class="t m0 x6 h3 y20 ff1 fs0 fc0 sc0 ls39 ws41">Re<span class="_0 blank"></span>vis<span class="_0 blank"></span>ão:<span class="_f blank"> </span>A<span class="_0 blank"></span>b o<span class="_0 blank"></span>rda<span class="_0 blank"></span>gem<span class="_13 blank"> </span>po<span class="_0 blank"></span>r<span class="_2 blank"> </span>L<span class="_0 blank"></span>inh<span class="_0 blank"></span>as</div><div class="t m0 x7 h4 yca ff1 fs1 fc1 sc0 ls39 ws1d">Dado um sistema de <span class="ff2 fc2 ls82">n</span><span class="ws42">equações e <span class="ff2 fc2 ls82">n</span><span class="ws2">va<span class="_0 blank"></span>riáveis, i.e.<span class="_2 blank"> </span>quando a matriz <span class="ff2">A</span></span></span></div><div class="t m0 x7 h4 ycb ff1 fs1 fc1 sc0 ls8f">é<span class="fc2 ls39 ws3">quadrada</span><span class="ls39 ws47">, mostramos que as seguintes a\u2026rmações são equivalentes:</span></div><div class="t m0 x2f h10 ycc ff2 fs5 fc0 sc0 ls6b">i<span class="ff7 ls6c">.</span><span class="fc1 ls33">A<span class="ff1 ls39 ws48">é não-singula<span class="_0 blank"></span>r.</span></span></div><div class="t m0 x32 h10 ycd ff2 fs5 fc0 sc0 ls39 ws3f">ii <span class="ff7 ls7c">.</span><span class="ff1 fc1 ws24">O sistema <span class="ff2 ws27">A<span class="ff6 ls34">x<span class="ff4 fs6 ls35">=</span><span class="ls90">b</span></span></span><span class="ws49">tem uma única solução pa<span class="_0 blank"></span>ra qualquer <span class="ff6 ws27">b<span class="ff7">.</span></span></span></span></div><div class="t m0 x3e h10 yce ff2 fs5 fc0 sc0 ls39 ws4a">iii <span class="ff7 ls91">.</span><span class="fc1 ls33">A<span class="ff1 ls39 ws4b">tem<span class="_c blank"> </span>p osto<span class="_c blank"> </span>máximo.</span></span></div><div class="t m0 x3e h10 ycf ff2 fs5 fc0 sc0 ls39 ws4c">iv <span class="ff7 ls92">.</span><span class="ff1 fc1 ws4d">det <span class="ff2 ls93">A<span class="ff3 fs6 ls94">6<span class="ff4 ls35">=</span></span></span><span class="ws27">0<span class="ff7">.</span></span></span></div><div class="t m0 x32 h10 yd0 ff2 fs5 fc0 sc0 ls95">v<span class="ff7 ls91">.</span><span class="fc1 ls33">A<span class="ff1 ls39 ws48">é invertível.</span></span></div><div class="t m0 x5 h5 y74 ff1 fs2 fc1 sc0 ls39">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/29c85f5a-8b02-49ce-9165-c9b1cf03d469/bg4.png"><div class="c x0 y1 w2 h2"><div class="t m0 x6 h3 y33 ff1 fs0 fc0 sc0 ls39 ws41">Re<span class="_0 blank"></span>vis<span class="_0 blank"></span>ão:<span class="_f blank"> </span>A<span class="_0 blank"></span>b o<span class="_0 blank"></span>rda<span class="_0 blank"></span>gem<span class="_13 blank"> </span>po<span class="_0 blank"></span>r<span class="_2 blank"> </span>C<span class="_0 blank"></span>olu<span class="_0 blank"></span>na<span class="_0 blank"></span>s</div><div class="t m0 x7 h2a yd1 ff1 fs1 fc1 sc0 ls39 ws4e">Dada uma matriz <span class="ff2 ls96">A<span class="ff4 fs3 ls4">=</span><span class="ff5 lsf va">\ue002</span></span><span class="ff6 ws3 v0">a</span><span class="fs2 ls10 v2">1</span><span class="ff7 ws9 v0">, ..., <span class="ff6 ws3">a<span class="ff2 fs2 ls97 v2">k</span><span class="ff5 ls98 va">\ue003</span></span></span><span class="ws2 v0">de tamanho <span class="ff2 ls99">n</span></span><span class="ff3 fs3 ls9a">\ue002</span><span class="ff2 ls9b">k</span><span class="ws1d">, de\u2026nimos</span></div><div class="t m0 x7 h2b yd2 ff1 fs1 fc1 sc0 ls9c">o<span class="fc2 ls39 wsd v0">espaço-coluna <span class="fc1 ws14">de <span class="ff2 ls9d">A</span><span class="ws1d">como o conjunto gerado p<span class="_4 blank"> </span>elas colunas de <span class="ff2 ws3">A</span>:</span></span></span></div><div class="t m0 xe h2c yd3 ff2 fs1 fc1 sc0 ls39 ws4f">Col <span class="ff4 fs3 ls15 v0">(</span><span class="ls16 v0">A<span class="ff4 fs3 ls17 v0">)<span class="ls18 v0">=<span class="ff3 ls19">L</span><span class="ls1a">[</span></span></span><span class="ff6 ls39 ws3">a<span class="ff1 fs2 ls1b v2">1</span><span class="ff7 ws10">, ..., </span>a</span><span class="fs2 ls1c v2">m</span><span class="ff4 fs3 ls39">]</span></span></div><div class="t m0 x7 h4 yd4 ff1 fs1 fc1 sc0 ls39 ws50">Neste caso, mostramos que:</div><div class="t m0 x2f h2d yd5 ff2 fs5 fc0 sc0 ls6b">i<span class="ff7 ls6c">.<span class="ff1 fc1 ls39 ws24">O sistema de equações <span class="ff2 ws27">A<span class="ff6 ls4d">x<span class="ff4 fs6 ls52">=</span><span class="ls9e">b</span></span></span><span class="ws29">tem uma solução para <span class="ff6 ls9f">b<span class="ff3 fs6 lsa0">2</span><span class="ff8 lsa1">R<span class="ff2 fs2 lsa2 vf">n</span></span></span><span class="ws51">se, e</span></span></span></span></div><div class="t m0 x15 h10 yd6 ff1 fs5 fc1 sc0 ls39 ws52">somente se, <span class="ff6 ls9f">b<span class="ff3 fs6 lsa3">2</span></span><span class="ff2 ws53">Col <span class="ff4 fs6 ls5c v0">(</span><span class="ls5d">A<span class="ff4 fs6 lsa4 v0">)</span></span><span class="ff7">.</span></span></div><div class="t m0 x32 h2e yd7 ff2 fs5 fc0 sc0 ls39 ws3f">ii <span class="ff7 ls7c">.</span><span class="ff1 fc1 ws24">O sistema de equações <span class="ff2 ws27">A<span class="ff6 ls4d">x<span class="ff4 fs6 ls52">=</span><span class="ls9e">b</span></span></span><span class="ws38">p ossui<span class="_c blank"> </span>solução<span class="_c blank"> </span>pa<span class="_0 blank"></span>ra<span class="_c blank"> </span>qualquer<span class="_c blank"> </span><span class="ff6 ls9f">b<span class="ff3 fs6 lsa0">2</span><span class="ff8 lsa5">R</span></span><span class="ff2 fs2 vf">n</span></span></span></div><div class="t m0 x15 h2f yd8 ff1 fs5 fc1 sc0 ls39 ws54">se, e somente se, <span class="ff2 ws39">Col <span class="ff4 fs6 ls5e v0">(</span><span class="ls5f">A<span class="ff4 fs6 ls60 v0">)<span class="ls35 v0">=</span></span><span class="ff8 lsa6">R</span><span class="fs2 lsa7 vf">n</span></span><span class="ff7">.</span></span></div><div class="t m0 x3f h5 yd9 ff1 fs2 fc1 sc0 ls39 ws3e">10</div></div><div class="c x0 y7 w2 h2"><div class="t m0 x6 h3 y8 ff1 fs0 fc0 sc0 ls39 ws55">T<span class="_15 blank"></span>ra<span class="_0 blank"></span>nfo<span class="_0 blank"></span>r<span class="_0 blank"></span>ma<span class="_0 blank"></span>ções L<span class="_0 blank"></span>ine<span class="_0 blank"></span>a<span class="_0 blank"></span>res</div><div class="t m0 x7 h4 yda ff1 fs1 fc1 sc0 ls39 ws1">Uma terceira fo<span class="_0 blank"></span>rma de ab<span class="_4 blank"> </span>orda<span class="_0 blank"></span>r um sistema de equações lineares é</div><div class="t m0 x7 hd ydb ff1 fs1 fc1 sc0 ls39 ws1d">interp<span class="_0 blank"></span>retá-lo como uma <span class="fc2 ws56 v0">função <span class="ff2 fc1 lsa8">T<span class="ffc lsa9">:<span class="ff8 lsaa">R</span></span><span class="fs2 lsab v4">k</span><span class="ff3 fs3 lsac">!</span><span class="ff8 lsad">R</span><span class="fs2 lsae v4">n</span></span><span class="fc1 ws2">, i.e.:</span></span></div><div class="t m0 x40 h7 ydc ff5 fs1 fc1 sc0 ls39">2</div><div class="t m0 x40 h7 ydd ff5 fs1 fc1 sc0 ls39">6</div><div class="t m0 x40 h7 yde ff5 fs1 fc1 sc0 ls39">6</div><div class="t m0 x40 h7 ydf ff5 fs1 fc1 sc0 ls39">6</div><div class="t m0 x40 h7 ye0 ff5 fs1 fc1 sc0 ls39">4</div><div class="t m0 x41 h30 ye1 ff2 fs1 fc1 sc0 ls5">a<span class="ff1 fs2 ls39 ws3e v2">11<span class="_5 blank"> </span></span><span class="ls6">a<span class="ff1 fs2 ls39 ws6 v2">12<span class="_5 blank"> </span></span><span class="ff3 fs3 ls39 ws7 v0">\ue001 \ue001 \ue001<span class="_6 blank"> </span></span><span class="v0">a<span class="ff1 fs2 ls7 v2">1<span class="ff2 ls39">k</span></span></span></span></div><div class="t m0 x41 h31 ye2 ff2 fs1 fc1 sc0 ls5">a<span class="ff1 fs2 ls39 ws3e v2">21<span class="_5 blank"> </span></span><span class="ls6">a<span class="ff1 fs2 ls39 ws6 v2">22<span class="_5 blank"> </span></span><span class="ff3 fs3 ls39 ws57 v0">\ue001 \ue001 \ue001<span class="_6 blank"> </span></span></span><span class="v0">a<span class="ff1 fs2 ls7 v2">2<span class="ff2 ls39">k</span></span></span></div><div class="t m0 x2d h4 ye3 ff1 fs1 fc1 sc0 ls39">.</div><div class="t m0 x2d h4 ye4 ff1 fs1 fc1 sc0 ls39">.</div><div class="t m0 x2d h32 ye5 ff1 fs1 fc1 sc0 lsaf">.<span class="ls39 v6">.</span></div><div class="t m0 x3 h4 ye6 ff1 fs1 fc1 sc0 ls39">.</div><div class="t m0 x3 h33 ye7 ff1 fs1 fc1 sc0 ls8">.<span class="lsb0 v3">.<span class="v16">.</span></span><span class="lsb1 v5">.</span><span class="ls39 v6">.</span></div><div class="t m0 x35 h4 ye8 ff1 fs1 fc1 sc0 ls39">.</div><div class="t m0 x35 h4 ye9 ff1 fs1 fc1 sc0 ls39">.</div><div class="t m0 x41 h34 yea ff2 fs1 fc1 sc0 lsb2">a<span class="fs2 lsc v2">n<span class="ff1 lsb3">1</span></span>a<span class="fs2 ls7d v2">n<span class="ff1 lsb4">2<span class="ff3 fs3 ls39 ws57 v7">\ue001 \ue001 \ue001<span class="_6 blank"> </span></span></span></span><span class="ls5 v0">a<span class="fs2 ls39 ws8 v2">nk</span></span></div><div class="t m0 x23 h7 yeb ff5 fs1 fc1 sc0 ls39">3</div><div class="t m0 x23 h7 yec ff5 fs1 fc1 sc0 ls39">7</div><div class="t m0 x23 h7 yed ff5 fs1 fc1 sc0 ls39">7</div><div class="t m0 x23 h7 yee ff5 fs1 fc1 sc0 ls39">7</div><div class="t m0 x23 h7 yef ff5 fs1 fc1 sc0 ls39">5</div><div class="t m0 x31 h7 yf0 ff5 fs1 fc1 sc0 ls39">2</div><div class="t m0 x31 h7 yf1 ff5 fs1 fc1 sc0 ls39">6</div><div class="t m0 x31 h7 yf2 ff5 fs1 fc1 sc0 ls39">6</div><div class="t m0 x31 h7 yf3 ff5 fs1 fc1 sc0 ls39">6</div><div class="t m0 x31 h7 yf4 ff5 fs1 fc1 sc0 ls39">4</div><div class="t m0 x42 h35 yf5 ff2 fs1 fc2 sc0 lsb5">x<span class="ff1 fs2 ls39 v2">1</span></div><div class="t m0 x42 h35 ye2 ff2 fs1 fc2 sc0 lsb5">x<span class="ff1 fs2 ls39 v2">2</span></div><div class="t m0 x43 h4 yf6 ff1 fs1 fc1 sc0 ls39">.</div><div class="t m0 x43 h4 yf7 ff1 fs1 fc1 sc0 ls39">.</div><div class="t m0 x43 h4 yf8 ff1 fs1 fc1 sc0 ls39">.</div><div class="t m0 x42 h35 yf9 ff2 fs1 fc2 sc0 lsb6">x<span class="fs2 ls39 v2">k</span></div><div class="t m0 x44 h7 yfa ff5 fs1 fc1 sc0 ls39">3</div><div class="t m0 x44 h7 yfb ff5 fs1 fc1 sc0 ls39">7</div><div class="t m0 x44 h7 yfc ff5 fs1 fc1 sc0 ls39">7</div><div class="t m0 x44 h7 yfd ff5 fs1 fc1 sc0 ls39">7</div><div class="t m0 x44 h36 yfe ff5 fs1 fc1 sc0 lsb7">5<span class="ff4 fs3 lsb8 v17">=</span><span class="ls39 v18">2</span></div><div class="t m0 x45 h7 yfb ff5 fs1 fc1 sc0 ls39">6</div><div class="t m0 x45 h7 yfc ff5 fs1 fc1 sc0 ls39">6</div><div class="t m0 x45 h7 yfd ff5 fs1 fc1 sc0 ls39">6</div><div class="t m0 x45 h7 yfe ff5 fs1 fc1 sc0 ls39">4</div><div class="t m0 x38 h35 yf5 ff2 fs1 fc3 sc0 lsb9">y<span class="ff1 fs2 ls39 v2">1</span></div><div class="t m0 x38 h35 ye2 ff2 fs1 fc3 sc0 lsb9">y<span class="ff1 fs2 ls39 v2">2</span></div><div class="t m0 x46 h4 yf6 ff1 fs1 fc1 sc0 ls39">.</div><div class="t m0 x46 h4 yf7 ff1 fs1 fc1 sc0 ls39">.</div><div class="t m0 x46 h4 yf8 ff1 fs1 fc1 sc0 ls39">.</div><div class="t m0 x38 h35 yf9 ff2 fs1 fc3 sc0 lsba">y<span class="fs2 ls39 v2">n</span></div><div class="t m0 x2b h7 yfa ff5 fs1 fc1 sc0 ls39">3</div><div class="t m0 x2b h7 yfb ff5 fs1 fc1 sc0 ls39">7</div><div class="t m0 x2b h7 yfc ff5 fs1 fc1 sc0 ls39">7</div><div class="t m0 x2b h7 yfd ff5 fs1 fc1 sc0 ls39">7</div><div class="t m0 x2b h7 yfe ff5 fs1 fc1 sc0 ls39">5</div><div class="t m0 x7 h37 yff ff1 fs1 fc1 sc0 ls39 ws58">A função <span class="ff2 lsbb">T<span class="ff4 fs3 ls29 v0">(</span><span class="ff6 lsbc v0">x<span class="ff4 fs3 lsbd v0">)<span class="lsbe v0">=</span></span><span class="ff2 ls39 ws3">A</span><span class="lsbf">x</span></span></span><span class="wsc v0">é chamada de <span class="fc2 ws1">transfo<span class="_0 blank"></span>rmação linear<span class="fc1">.</span></span></span></div><div class="t m0 x7 h4 y100 ff1 fs1 fc1 sc0 ls39 ws1">Uma matriz qualquer determina uma <span class="fc2 ws59">única </span><span class="ws17">transfo<span class="_0 blank"></span>rmação linear.</span></div><div class="t m0 x7 h4 y101 ff1 fs1 fc1 sc0 ls39 ws1">Uma transfo<span class="_0 blank"></span>rmação linear possui as seguintes prop<span class="_0 blank"></span>riedades:</div><div class="t m0 x2f h23 y102 ff2 fs5 fc0 sc0 ls6b">i<span class="ff7 ls6c">.</span><span class="fc1 lsc0">T<span class="ff4 fs6 ls5e v0">(</span><span class="ff6 lsc1">x<span class="ff4 fs6 lsc2 v0">)<span class="ls6a v0">+</span></span></span><span class="lsc3">T<span class="ff4 fs6 ls61 v0">(</span><span class="ff6 lsc4">y<span class="ff4 fs6 ls60 v0">)<span class="ls35 v0">=</span></span></span></span>T<span class="ff4 fs6 ls5e v0">(</span><span class="ff6 lsc5">x<span class="ff4 fs6 ls7a">+</span><span class="lsc6">y<span class="ff4 fs6 ls39 v0">)</span></span></span></span></div><div class="t m0 x32 h23 y103 ff2 fs5 fc0 sc0 ls39 ws3f">ii <span class="ff7 ls7c">.</span><span class="fc1 lsc0">T<span class="ff4 fs6 ls5e v0">(</span><span class="lsc7">c<span class="ff6 lsc8">x<span class="ff4 fs6 lsc9 v0">)<span class="ls52 v0">=</span></span></span><span class="ls39 ws5a">cT <span class="ff4 fs6 ls5c v0">(</span><span class="ff6 lsca">x</span><span class="ff4 fs6 v0">)</span></span></span></span></div><div class="t m0 x3f h5 y104 ff1 fs2 fc1 sc0 ls39 ws3e">11</div></div><div class="c x0 y1f w2 h2"><div class="t m0 x6 h3 y20 ff1 fs0 fc0 sc0 ls39 ws55">T<span class="_15 blank"></span>ra<span class="_0 blank"></span>nfo<span class="_0 blank"></span>r<span class="_0 blank"></span>ma<span class="_0 blank"></span>ções L<span class="_0 blank"></span>ine<span class="_0 blank"></span>a<span class="_0 blank"></span>res</div><div class="t m0 x7 h4 y105 ff1 fs1 fc1 sc0 ls39 ws5b">Dada uma <span class="fc2 ws5c">transfo<span class="_0 blank"></span>rmação linear</span></div><div class="t m0 x47 h38 y106 ff2 fs1 fc1 sc0 lscb">T<span class="ffc lscc">:<span class="ff8 lscd v0">R</span></span><span class="fs2 ls39 v4">k</span></div><div class="t m0 x10 h39 y107 ff5 fs1 fc1 sc0 lsce">|<span class="ls39 ws5d v0">{<span class="_4 blank"></span>z }</span></div><div class="t m0 x16 h5 y108 ff1 fs2 fc1 sc0 ls39 ws5e">D o<span class="_4 blank"> </span>m<span class="_16 blank"> </span>ín i o</div><div class="t m0 x35 h3a y106 ff3 fs3 fc1 sc0 lscf">!<span class="ff8 fs1 lsad v0">R<span class="ff2 fs2 ls39 v4">n</span></span></div><div class="t m0 x42 h7 y109 ff5 fs1 fc1 sc0 lsd0">|<span class="ls39 ws5f v0">{<span class="_4 blank"></span>z }</span></div><div class="t m0 x25 h5 y10a ff1 fs2 fc1 sc0 ls39 ws3e">C o n tr a - d o m í n io</div><div class="t m0 x7 h4 y10b ff1 fs1 fc1 sc0 ls39 ws3">de\u2026nimos:</div><div class="t m0 x2f h10 y10c ff2 fs5 fc0 sc0 ls6b">i<span class="ff7 ls6c">.<span class="ff1 fc2 ls39 ws25">Imagem de T:</span></span></div><div class="t m0 x33 h3b y10d ffc fs5 fc1 sc0 ls39 ws60">Im <span class="ff4 fs6 ls61 v0">(</span><span class="ff2 lsd1">T<span class="ff4 fs6 ls60 v0">)<span class="ls37 v0">=<span class="ff3 lsd2">f</span></span></span><span class="ff6 lsd3">y<span class="ff3 fs6 lsd4">2</span><span class="ff8 lsa1">R</span></span><span class="fs2 lsd5 v4">n</span></span><span class="lsd6">:<span class="ff2 lsc0">T<span class="ff4 fs6 ls5e v0">(</span><span class="ff6 lsc1">x<span class="ff4 fs6 ls60 v0">)<span class="ls35 v0">=</span></span><span class="lsd7">y</span></span></span></span><span class="ff1 ws24">pa<span class="_0 blank"></span>ra algum <span class="ff6 ls3a">x<span class="ff3 fs6 lsa0">2</span><span class="ff8 lsa1">R<span class="ff2 fs2 lsd8 v4">k</span></span></span><span class="ff3 fs6">g</span></span></div><div class="t m0 x32 h10 y10e ff2 fs5 fc0 sc0 ls39 ws3f">ii <span class="ff7 ls7c">.</span><span class="ff1 fc2 ws25">Núcleo de T:</span></div><div class="t m0 x3 h20 y10f ff2 fs5 fc1 sc0 ls39 ws36">Nul <span class="ff4 fs6 ls5e v0">(</span><span class="lsd1">T<span class="ff4 fs6 ls60 v0">)<span class="lsd9 v0">=<span class="ff3 lsd2">f</span></span></span><span class="ff6 ls3a">x<span class="ff3 fs6 lsa0">2</span><span class="ff8 lsa6">R</span></span><span class="fs2 ls76 v4">k</span><span class="ffc lsda">:</span><span class="lsc0">T<span class="ff4 fs6 ls5e v0">(</span><span class="ff6 ls5a">x<span class="ff4 fs6 ls60 v0">)<span class="ls52 v0">=</span></span><span class="lsdb">0</span></span></span></span><span class="ff3 fs6">g</span></div><div class="t m0 x3f h5 y74 ff1 fs2 fc1 sc0 ls39 ws3e">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/29c85f5a-8b02-49ce-9165-c9b1cf03d469/bg5.png"><div class="c x0 y1 w2 h2"><div class="t m0 x6 h3 y33 ff1 fs0 fc0 sc0 ls39 ws61">T<span class="_15 blank"></span>eo<span class="_3 blank"></span>rem<span class="_0 blank"></span>a F<span class="_0 blank"></span>un<span class="_0 blank"></span>dam<span class="_0 blank"></span>en<span class="_0 blank"></span>tal d<span class="_0 blank"></span>a Álg<span class="_0 blank"></span>eb<span class="_0 blank"></span>r<span class="_0 blank"></span>a</div><div class="t m0 x7 h4 y110 ff1 fs1 fc2 sc0 ls39 ws62">T<span class="_3 blank"></span>eo<span class="_0 blank"></span>rema: <span class="fc1 ws63">Dada uma transformação linea<span class="_0 blank"></span>r <span class="ff2 lsa8">T<span class="ffc lsdc">:<span class="ff8 lsdd">R</span></span><span class="fs2 lsde v4">k</span><span class="ff3 fs3 lsac">!</span><span class="ff8 lsdd">R</span><span class="fs2 lsdf v4">n</span></span><span class="ws2">, temos:</span></span></div><div class="t m0 x48 h3c y111 ff1 fs1 fc1 sc0 ls39 ws18">dim <span class="ff8 lsad">R<span class="ff2 fs2 lse0 v19">k</span><span class="ff4 fs3 lse1 v0">=</span></span><span class="ws64 v0">dim <span class="ff2 ws65">Nul <span class="ff4 fs3 ls15">(</span><span class="lse2">T</span><span class="ff4 fs3 ws66">) + </span></span><span class="ws67">dim <span class="ffc ws68">Im <span class="ff4 fs3 ls29 v0">(</span><span class="ff2 lse3">T<span class="ff4 fs3 lse4 v0">)</span></span><span class="ff7">,</span></span></span></span></div><div class="t m0 x7 h3d y112 ff1 fs1 fc1 sc0 ls39 ws69">onde<span class="_7 blank"> </span>dim <span class="ff8 ls12">R<span class="ff2 fs2 lse5 v4">k</span><span class="ff4 fs3 ls1e v0">=<span class="ff2 fs1 ls9b">k</span></span></span><span class="v0">.</span></div><div class="t m0 x3f h5 y6 ff1 fs2 fc1 sc0 ls39 ws3e">13</div></div><div class="c x0 y7 w2 h2"><div class="t m0 x6 h3 y8 ff1 fs0 fc0 sc0 ls39 ws55">T<span class="_15 blank"></span>ra<span class="_0 blank"></span>nfo<span class="_0 blank"></span>r<span class="_0 blank"></span>ma<span class="_0 blank"></span>ções L<span class="_0 blank"></span>ine<span class="_0 blank"></span>a<span class="_0 blank"></span>res</div><div class="t m0 x7 h4 y113 ff1 fs1 fc2 sc0 ls39 ws6a">De\u2026nição: <span class="fc1 ws1">Uma transfo<span class="_0 blank"></span>rmação linear <span class="ff2 lscb">T<span class="ffc lse6">:<span class="ff8 ls12">R</span></span><span class="fs2 lse7 v4">k</span><span class="ff3 fs3 lse8">!</span><span class="ff8 ls12">R</span><span class="fs2 lse9 v4">n</span></span><span class="lsea">é</span><span class="fc2 ws6b">injeto<span class="_0 blank"></span>ra <span class="fc1 ws1d">se, e</span></span></span></div><div class="t m0 x7 h4 y114 ff1 fs1 fc1 sc0 ls39 ws6c">somente se, pa<span class="_0 blank"></span>ra quaisquer <span class="ff6 ws3">x</span><span class="fs2 lseb v2">1</span><span class="ff7 lsec">,</span><span class="ff6 ws3">x</span><span class="fs2 lsed v2">2</span><span class="ff3 fs3 lsee v0">2<span class="ff8 fs1 lscd">R<span class="ff2 fs2 lsef v4">k</span></span></span><span class="v0">:</span></div><div class="t m0 x49 h3e y115 ff2 fs1 fc1 sc0 lsf0">T<span class="ff4 fs3 ls21 v0">(</span><span class="ff6 ls39 ws3 v0">x<span class="ff1 fs2 lsf1 v2">1</span><span class="ff4 fs3 ls17 v0">)<span class="lsf2 v0">=</span></span><span class="ff2 lsf3">T<span class="ff4 fs3 ls15 v0">(</span></span>x<span class="ff1 fs2 lsf1 v2">2</span><span class="ff4 fs3 lsf4 v0">)<span class="ff3 lsf5 v0">)</span></span>x<span class="ff1 fs2 lsf6 v2">1</span><span class="ff4 fs3 ls1e">=</span>x<span class="ff1 fs2 v2">2</span></span></div><div class="t m0 x7 h4 y116 ff1 fs1 fc2 sc0 ls39 ws6d">De\u2026nição: <span class="fc1 ws6e">Uma transfo<span class="_0 blank"></span>rmação linear <span class="ff2 lsf7">T<span class="ffc lsf8">:<span class="ff8 lsad">R</span></span><span class="fs2 lsf9 v4">k</span><span class="ff3 fs3 lse8">!</span><span class="ff8 lscd">R</span><span class="fs2 lsfa v4">n</span></span><span class="lsfb">é</span><span class="fc2 ws6f">sob<span class="_0 blank"></span>rejetora <span class="fc1 ws70">se, e</span></span></span></div><div class="t m0 x7 h4 y117 ff1 fs1 fc1 sc0 ls39 ws71">somente se, <span class="ffc ws72">Im <span class="ff4 fs3 ls21 v0">(</span><span class="ff2 lse2 v0">T<span class="ff4 fs3 lsbd v0">)<span class="lsbe v0">=</span></span><span class="ff8 lsad">R</span><span class="fs2 lsfc v4">n</span><span class="ff7 ls39">.</span></span></span></div><div class="t m0 x3f h5 yc9 ff1 fs2 fc1 sc0 ls39 ws3e">14</div></div><div class="c x0 y1f w2 h2"><div class="t m0 x6 h3 y20 ff1 fs0 fc0 sc0 ls39 ws55">T<span class="_15 blank"></span>ra<span class="_0 blank"></span>nfo<span class="_0 blank"></span>r<span class="_0 blank"></span>ma<span class="_0 blank"></span>ções L<span class="_0 blank"></span>ine<span class="_0 blank"></span>a<span class="_0 blank"></span>res</div><div class="t m0 x7 h4 y118 ff1 fs1 fc2 sc0 ls39 ws73">T<span class="_3 blank"></span>eo<span class="_0 blank"></span>rema: <span class="fc1 ws1">Uma transformação linea<span class="_0 blank"></span>r <span class="ff2 lscb">T<span class="ffc lsa9">:<span class="ff8 lsaa">R</span></span><span class="fs2 lsfd v4">k</span><span class="ff3 fs3 lsac">!</span><span class="ff8 lsad">R</span><span class="fs2 lsfe v4">n</span></span><span class="ws2">é injetora se, e</span></span></div><div class="t m0 x7 h4 y119 ff1 fs1 fc1 sc0 ls39 ws74">somente se, <span class="ff2 ws75">Nul <span class="ff4 fs3 ls15 v0">(</span><span class="lse2 v0">T<span class="ff4 fs3 lsbd v0">)<span class="ls18 v0">=<span class="ff3 lsff v0">f</span></span></span><span class="ff6 ls100">0<span class="ff3 fs3 ls101 v0">g</span><span class="ff7 ls39">.</span></span></span></span></div><div class="t m0 x7 h4 y11a ff1 fs1 fc2 sc0 ls39 ws3">Prova:</div><div class="t m0 x4a h3f y11b ff4 fs6 fc0 sc0 ls102">(<span class="ff3 ls103">(</span><span class="ls104">)<span class="ff1 fs5 fc1 ls39 ws2d">Sup onha<span class="_c blank"> </span>que<span class="_c blank"> </span><span class="ff2 ws33">Nul </span></span><span class="fc1 ls5e v0">(<span class="ff2 fs5 lsd1 v0">T</span><span class="ls60">)<span class="ls37 v0">=<span class="ff3 ls105">f<span class="ff6 fs5 lsdb">0</span><span class="ls106">g<span class="ff1 fs5 ls39 ws76">e considere <span class="ff6 ws27">x</span><span class="fs2 ls10 v2">1</span><span class="ff7 ls107">,</span><span class="ff6 ws27">x</span><span class="fs2 ls108 v2">2</span></span><span class="lsa0">2<span class="ff8 fs5 lsa1">R<span class="ff2 fs2 ls109 vf">k</span><span class="ff1 ls39 ws29">quaisquer com</span></span></span></span></span></span></span></span></span></div><div class="t m0 x4b h40 y11c ff2 fs5 fc1 sc0 ls10a">T<span class="ff4 fs6 ls61 v0">(</span><span class="ff6 ls39 ws27">x<span class="ff1 fs2 ls10b v2">1</span><span class="ff4 fs6 ls60 v0">)<span class="ls35 v0">=</span></span></span><span class="ls10c v0">T<span class="ff4 fs6 ls10d v0">(</span><span class="ff6 ls39 ws27">x<span class="ff1 fs2 ls10e v2">2</span><span class="ff4 fs6 v0">)</span></span></span></div><div class="t m0 x15 h10 y11d ff1 fs5 fc1 sc0 ls39 ws25">Note que p<span class="_4 blank"> </span>odemos re-escrever a expressão acima como:</div><div class="t m0 x8 h40 y11e ff2 fs5 fc1 sc0 ls10a">T<span class="ff4 fs6 ls5c v0">(</span><span class="ff6 ls39 ws27">x<span class="ff1 fs2 ls10e v2">1</span><span class="ff4 fs6 lsc9 v0">)<span class="ls52 v0">=</span></span></span><span class="lsc0 v0">T<span class="ff4 fs6 ls5e v0">(</span><span class="ff6 ls39 ws27">x<span class="ff1 fs2 ls10b v2">2</span><span class="ff4 fs6 ls10f v0">)<span class="ff3 ls110 v0">$</span></span><span class="ff2 v0">A<span class="ff6">x<span class="ff1 fs2 ls108 v2">1</span><span class="ff4 fs6 ls52">=</span></span>A<span class="ff6">x<span class="ff1 fs2 ls111 v2">2</span><span class="ff3 fs6 ls110">$</span></span><span class="ls5f">A<span class="ff4 fs6 ls5e">(</span></span><span class="ff6">x<span class="ff1 fs2 ls6e v2">1</span><span class="ff3 fs6 ls6a">\ue000</span>x<span class="ff1 fs2 ls112 v2">2</span><span class="ff4 fs6 ws77">) = </span>0</span></span></span></span></div><div class="t m0 x4c h23 y11f ff3 fs6 fc1 sc0 ls113">$<span class="ff2 fs5 lsd1">T</span><span class="ff4 ls5e">(<span class="ff6 fs5 ls39 ws27">x<span class="ff1 fs2 ls70 v2">1</span></span></span><span class="ls71">\ue000<span class="ff6 fs5 ls39 ws27">x<span class="ff1 fs2 ls10b v2">2</span></span><span class="ff4 ls39 ws77">) = <span class="ff6 fs5">0</span></span></span></div><div class="t m0 x15 h10 y120 ff1 fs5 fc1 sc0 ls39 ws78">Assim, como <span class="ff2 ws33">Nul <span class="ff4 fs6 ls114 v0">(</span><span class="ls115">T<span class="ff4 fs6 lsc9 v0">)<span class="ls116 v0">=<span class="ff3 lsd2">f</span></span></span><span class="ff6 lsdb">0<span class="ff3 fs6 lsd2">g</span></span></span></span><span class="ws25">, segue que:</span></div><div class="t m0 x4d h23 y121 ff6 fs5 fc1 sc0 ls39 ws27">x<span class="ff1 fs2 ls6e v2">1</span><span class="ff3 fs6 ls7a">\ue000</span>x<span class="ff1 fs2 ls117 v2">2</span><span class="ff4 fs6 ls52">=</span><span class="ls118">0<span class="ff3 fs6 ls119">)</span></span>x<span class="ff1 fs2 ls117 v2">1</span><span class="ff4 fs6 ls35">=</span>x<span class="ff1 fs2 v2">2</span></div><div class="t m0 x15 h10 y122 ff1 fs5 fc1 sc0 ls39 ws79">P<span class="_0 blank"></span>ortanto, <span class="ff2 ls11a">T</span><span class="ws48">é injeto<span class="_0 blank"></span>ra.</span></div><div class="t m0 x3f h5 y123 ff1 fs2 fc1 sc0 ls39 ws3e">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/29c85f5a-8b02-49ce-9165-c9b1cf03d469/bg6.png"><div class="c x0 y1 w2 h2"><div class="t m0 x6 h3 y33 ff1 fs0 fc0 sc0 ls39 ws55">T<span class="_15 blank"></span>ra<span class="_0 blank"></span>nfo<span class="_0 blank"></span>r<span class="_0 blank"></span>ma<span class="_0 blank"></span>ções L<span class="_0 blank"></span>ine<span class="_0 blank"></span>a<span class="_0 blank"></span>res</div><div class="t m0 x7 h4 y124 ff1 fs1 fc1 sc0 ls39 ws3">(Cont.)</div><div class="t m0 x4a h41 y125 ff4 fs6 fc0 sc0 ls102">(<span class="ff3 ls103">)</span><span class="ls104">)<span class="ff1 fs5 fc1 ls39 ws2d">Sup onha<span class="_c blank"> </span>que<span class="_c blank"> </span><span class="ff2 ls11b">T</span><span class="ws7a">é injeto<span class="_0 blank"></span>ra e considere <span class="ff6 ls34">x<span class="ff3 fs6 lsd4">2</span><span class="ff8 ls11c">R<span class="ff2 fs2 ls11d vf">k</span></span></span><span class="ws7b">qualquer com</span></span></span></span></div><div class="t m0 x4e h10 y126 ff2 fs5 fc1 sc0 lsc0">T<span class="ff4 fs6 ls5e v0">(</span><span class="ff6 lsc1">x<span class="ff4 fs6 lsc9 v0">)<span class="ls52 v0">=</span></span><span class="ff1 ls39">0</span></span></div><div class="t m0 x15 h10 y127 ff1 fs5 fc1 sc0 ls39 ws2d">Note<span class="_c blank"> </span>que<span class="_c blank"> </span>po demos<span class="_c blank"> </span>escrever:</div><div class="t m0 x17 h23 y128 ff2 fs5 fc1 sc0 lsc0">T<span class="ff4 fs6 ls61 v0">(</span><span class="ff6 ls5a">x<span class="ff4 fs6 ls60 v0">)<span class="ls35 v0">=</span></span></span><span class="ls10a">T<span class="ff4 fs6 ls114 v0">(</span><span class="ff6 ls11e">0<span class="ff4 fs6 ls39 v0">)</span></span></span></div><div class="t m0 x15 h10 y129 ff1 fs5 fc1 sc0 ls39 ws7c">Assim, como <span class="ff2 ls11b">T</span><span class="ws30">é injeto<span class="_0 blank"></span>ra, temos que:</span></div><div class="t m0 x4f h23 y12a ff6 fs5 fc1 sc0 ls3a">x<span class="ff4 fs6 ls35">=</span><span class="ls39">0</span></div><div class="t m0 x15 h10 y12b ff1 fs5 fc1 sc0 ls39 ws7d">P<span class="_0 blank"></span>ortanto, <span class="ff2 ws7e">Nul <span class="ff4 fs6 ls61">(</span><span class="lsd1">T</span><span class="ff4 fs6 ws7f">) = <span class="ff3 ls11f">f</span></span><span class="ff6 lsdb">0<span class="ff3 fs6 ls105">g</span></span><span class="ff7">.</span></span></div><div class="t m0 x3f h5 y12c ff1 fs2 fc1 sc0 ls39 ws3e">16</div></div><div class="c x0 y7 w2 h2"><div class="t m0 x6 h3 y8 ff1 fs0 fc0 sc0 ls39 ws55">T<span class="_15 blank"></span>ra<span class="_0 blank"></span>nfo<span class="_0 blank"></span>r<span class="_0 blank"></span>ma<span class="_0 blank"></span>ções L<span class="_0 blank"></span>ine<span class="_0 blank"></span>a<span class="_0 blank"></span>res</div><div class="t m0 x7 h42 y12d ff1 fs1 fc2 sc0 ls39 ws1d">Exemplo 1:<span class="_2 blank"> </span><span class="fc1 ws80">Considere a transfo<span class="_0 blank"></span>rmação linear <span class="ff2 lsa8">T<span class="ffc lsa9">:<span class="ff8 lsaa">R</span></span></span><span class="fs2 ls120 v4">3</span><span class="ff3 fs3 lsac">!</span><span class="ff8 lsdd">R</span><span class="fs2 ls121 v4">2</span><span class="ws2">, com:</span></span></div><div class="t m0 x50 h43 y12e ff2 fs1 fc1 sc0 ls122">T<span class="ff4 fs3 ls15 v0">(</span><span class="ff6 lsbc v0">x<span class="ff4 fs3 ls123 v0">)<span class="ls124 v0">=</span></span><span class="ff5 ls125 v1a">\ue014</span><span class="ff1 ls39 ws23 v3">1 0 0</span></span></div><div class="t m0 x0 h44 y12f ff1 fs1 fc1 sc0 ls39 ws81">0<span class="_a blank"> </span>1<span class="_a blank"> </span>0 <span class="ff5 ls126 vb">\ue015</span><span class="ff5 v1b">2</span></div><div class="t m0 x17 h7 y130 ff5 fs1 fc1 sc0 ls39">4</div><div class="t m0 x35 h35 y131 ff2 fs1 fc1 sc0 ls39 ws3">x<span class="ff1 fs2 v2">1</span></div><div class="t m0 x35 h35 y132 ff2 fs1 fc1 sc0 ls39 ws3">x<span class="ff1 fs2 v2">2</span></div><div class="t m0 x35 h35 y133 ff2 fs1 fc1 sc0 ls39 ws3">x<span class="ff1 fs2 v2">3</span></div><div class="t m0 x26 h7 y134 ff5 fs1 fc1 sc0 ls39">3</div><div class="t m0 x26 h45 y135 ff5 fs1 fc1 sc0 ls127">5<span class="ff3 fs3 ls128 v13">)</span><span class="ls125 v1c">\ue014</span><span class="ff2 ls39 ws3 v19">y<span class="ff1 fs2 v2">1</span></span></div><div class="t m0 x13 h46 y136 ff2 fs1 fc1 sc0 ls39 ws3">y<span class="ff1 fs2 ls129 v2">2</span><span class="ff5 ls12a vb">\ue015</span><span class="ff4 fs3 lsb8 v3">=</span><span class="ff5 ls12b vb">\ue014</span><span class="ls12c v1d">x</span><span class="ff1 fs2 v1e">1</span></div><div class="t m0 x51 h47 y136 ff2 fs1 fc1 sc0 ls12c">x<span class="ff1 fs2 ls12d v2">2</span><span class="ff5 ls39 vb">\ue015</span></div><div class="t m0 x14 h48 y137 ff9 fs4 fc0 sc0 ls32">I<span class="ff1 fs5 fc1 ls39 ws48 v2">Note que:</span></div><div class="t m0 x11 h10 y138 ff1 fs5 fc1 sc0 ls39 ws82">dim <span class="ffc ws83">Im <span class="ff4 fs6 ls5e v0">(</span><span class="ff2 ls115">T<span class="ff4 fs6 ls60 v0">)<span class="ls35 v0">=</span></span></span></span>2</div><div class="t m0 x15 h49 y139 ff1 fs5 fc1 sc0 ls39 ws79">P<span class="_0 blank"></span>ortanto, <span class="ffc ws83">Im <span class="ff4 fs6 ls5e v0">(</span><span class="ff2 ls12e">T<span class="ff4 fs6 ls60 v0">)<span class="ls12f v0">=</span></span><span class="ff8 lsa5">R</span></span></span><span class="fs2 ls1b vf">2</span><span class="ws25">, o que implica que <span class="ff2 ls130">T</span><span class="ls131">é</span><span class="fc2 ws27 v0">sob<span class="_0 blank"></span>rejetora<span class="fc1">.</span></span></span></div><div class="t m0 x14 hf y13a ff9 fs4 fc0 sc0 ls32">I<span class="ff1 fs5 fc1 ls39 ws30 v2">P<span class="_0 blank"></span>elo teorema fundamental da álgeb<span class="_0 blank"></span>ra, segue que:</span></div><div class="t m0 x4d h10 y13b ff1 fs5 fc1 sc0 ls39 ws84">dim <span class="ff2 ws36">Nul <span class="ff4 fs6 ls5e v0">(</span><span class="lsd1">T<span class="ff4 fs6 ls60 v0">)<span class="ls35 v0">=</span></span></span></span><span class="ls69">3<span class="ff3 fs6 ls132">\ue000</span><span class="ls3a">2<span class="ff4 fs6 ls52">=</span></span></span>1</div><div class="t m0 x15 h4a y13c ff1 fs5 fc1 sc0 ls39 ws85">Assim, <span class="ff2 ws33">Nul <span class="ff4 fs6 ls5e v0">(</span><span class="lsd1">T<span class="ff4 fs6 ls60 v0">)<span class="ff3 ls133 v0">6<span class="ff4 ls37">=</span><span class="ls11f">f</span></span></span><span class="ff6 lsdb">0<span class="ff3 fs6 ls105">g</span></span></span></span><span class="ws25">, o que implica que <span class="ff2 ls134">T</span><span class="fc2 ws34 v0">não é injeto<span class="_0 blank"></span>ra<span class="fc1">.</span></span></span></div><div class="t m0 x3f h5 yc9 ff1 fs2 fc1 sc0 ls39 ws3e">17</div></div><div class="c x0 y1f w2 h2"><div class="t m0 x6 h3 y20 ff1 fs0 fc0 sc0 ls39 ws55">T<span class="_15 blank"></span>ra<span class="_0 blank"></span>nfo<span class="_0 blank"></span>r<span class="_0 blank"></span>ma<span class="_0 blank"></span>ções L<span class="_0 blank"></span>ine<span class="_0 blank"></span>a<span class="_0 blank"></span>res</div><div class="t m0 x7 h42 y13d ff1 fs1 fc2 sc0 ls39 ws1d">Exemplo 2:<span class="_2 blank"> </span><span class="fc1 ws80">Considere a transfo<span class="_0 blank"></span>rmação linear <span class="ff2 lsa8">T<span class="ffc lsa9">:<span class="ff8 lsaa">R</span></span></span><span class="fs2 ls120 v4">2</span><span class="ff3 fs3 lsac">!</span><span class="ff8 lsdd">R</span><span class="fs2 ls121 v4">2</span><span class="ws2">, com:</span></span></div><div class="t m0 x52 h4b y13e ff2 fs1 fc1 sc0 ls122">T<span class="ff4 fs3 ls15 v0">(</span><span class="ff6 lsbc v0">x<span class="ff4 fs3 ls123 v0">)<span class="ls124 v0">=</span></span><span class="ff5 ls135 v1a">\ue014</span><span class="ff1 ls39 ws86 v3">1 3</span></span></div><div class="t m0 xf h4c y13f ff1 fs1 fc1 sc0 ls39 ws87">2<span class="_a blank"> </span>1 <span class="ff5 ws88 vb">\ue015 \ue014<span class="_b blank"> </span></span><span class="ff2 ls12c v1d">x</span><span class="fs2 v1e">1</span></div><div class="t m0 x4c h4d y140 ff2 fs1 fc1 sc0 ls12c">x<span class="ff1 fs2 ls136 v2">2</span><span class="ff5 ls137 vb">\ue015</span><span class="ff3 fs3 ls138 v3">)</span><span class="ff5 ls125 vb">\ue014</span><span class="ls39 ws3 v1d">y<span class="ff1 fs2 v2">1</span></span></div><div class="t m0 x43 h4d y141 ff2 fs1 fc1 sc0 ls39 ws3">y<span class="ff1 fs2 ls129 v2">2</span><span class="ff5 ls139 vb">\ue015</span><span class="ff4 fs3 ls4 v3">=</span><span class="ff5 ls125 vb">\ue014</span><span class="ls12c v1d">x</span><span class="ff1 fs2 ls13a v1e">1</span><span class="ff4 fs3 ls13b v1d">+</span><span class="ff1 v1d">2</span><span class="ls13c v1d">x</span><span class="ff1 fs2 v1e">2</span></div><div class="t m0 x2a h4e y13f ff1 fs1 fc1 sc0 ls39 ws3">2<span class="ff2 ls12c">x</span><span class="fs2 ls13d v2">1</span><span class="ff4 fs3 ls9a v0">+<span class="ff2 fs1 ls13c">x</span></span><span class="fs2 ls13e v2">2</span><span class="ff5 vb">\ue015</span></div><div class="t m0 x14 h48 y142 ff9 fs4 fc0 sc0 ls32">I<span class="ff1 fs5 fc1 ls39 ws48 v2">Note que:</span></div><div class="t m0 x11 h10 y143 ff1 fs5 fc1 sc0 ls39 ws82">dim <span class="ffc ws83">Im <span class="ff4 fs6 ls5e v0">(</span><span class="ff2 ls115">T<span class="ff4 fs6 ls60 v0">)<span class="ls35 v0">=</span></span></span></span>2</div><div class="t m0 x15 h49 y144 ff1 fs5 fc1 sc0 ls39 ws79">P<span class="_0 blank"></span>ortanto, <span class="ffc ws83">Im <span class="ff4 fs6 ls5e v0">(</span><span class="ff2 ls12e">T<span class="ff4 fs6 ls60 v0">)<span class="ls12f v0">=</span></span><span class="ff8 lsa5">R</span></span></span><span class="fs2 ls1b vf">2</span><span class="ws25">, o que implica que <span class="ff2 ls130">T</span><span class="ls131">é</span><span class="fc2 ws27 v0">sob<span class="_0 blank"></span>rejetora<span class="fc1">.</span></span></span></div><div class="t m0 x14 hf y145 ff9 fs4 fc0 sc0 ls32">I<span class="ff1 fs5 fc1 ls39 ws30 v2">P<span class="_0 blank"></span>elo teorema fundamental da álgeb<span class="_0 blank"></span>ra, segue que:</span></div><div class="t m0 x4d h10 y146 ff1 fs5 fc1 sc0 ls39 ws84">dim <span class="ff2 ws36">Nul <span class="ff4 fs6 ls5e v0">(</span><span class="lsd1">T<span class="ff4 fs6 ls60 v0">)<span class="ls35 v0">=</span></span></span></span><span class="ls69">2<span class="ff3 fs6 ls132">\ue000</span><span class="ls3a">2<span class="ff4 fs6 ls52">=</span></span></span>0</div><div class="t m0 x15 h10 y147 ff1 fs5 fc1 sc0 ls39 ws85">Assim, <span class="ff2 ws33">Nul <span class="ff4 fs6 ls5e v0">(</span><span class="lsd1">T<span class="ff4 fs6 ls60 v0">)<span class="ls50 v0">=<span class="ff3 ls11f">f</span></span></span><span class="ff6 lsdb">0<span class="ff3 fs6 lsd2">g</span></span></span></span><span class="ws30">, o que implica que <span class="ff2 ls130">T</span><span class="ls13f">é</span><span class="fc2 ws27 v0">injeto<span class="_0 blank"></span>ra<span class="fc1">.</span></span></span></div><div class="t m0 x14 h4f y148 ff9 fs4 fc0 sc0 ls32">I<span class="ff1 fs5 fc1 ls39 ws25 v2">Se uma função é tanto injeto<span class="_0 blank"></span>ra quanto sobrejeto<span class="_0 blank"></span>ra, então dizemos que</span></div><div class="t m0 x15 h10 y149 ff1 fs5 fc1 sc0 ls39 ws89">ela é <span class="fc2 ws27">bijeto<span class="_0 blank"></span>ra<span class="fc1">.</span></span></div><div class="t m0 x3f h5 y74 ff1 fs2 fc1 sc0 ls39 ws3e">18</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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