Rahasia Ilmu Eksakta

Soal dan jawaban Anril(analisis real) 2.1

<div dir="ltr" style="text-align: left;" trbidi="on"><div style="text-align: justify;"><span style="font-family: "times" , "times new roman" , serif;">Untuk membantu anda keluar dari jerat kesulitan mengerjakan soal analisis real 1, saya akan membantu dengan postingan soal analisi real 1 yaitu pembahasan soal latihan 2.1 yang terdapat dalam buku karangan Bartle Dan Sherbert. Mari disimak teman-teman. </span></div><div style="text-align: justify;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></div><div style="text-align: justify;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></div><div style="text-align: justify;"><span style="font-family: "times" , "times new roman" , serif;"><b>Untuk soal no 1</b></span></div><div style="text-align: justify;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg-Bu25vYnpjz5zu59hNNVdHERMnRGh-4PQ4QYoPkS_WzuivIoPcfsI2pw6_-05pjzEDRGcEsX4Mrqube-F5e4WQ_IzznzblvxPSDjKNI8vKH2IoX1rwIKWnu58A-f4uovSjd5O0QXqXf0/s1600/soal+no+1+analisis+real+bartle+2.1.jpg"><span style="font-family: "times" , "times new roman" , serif;"><img border="0" height="44" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg-Bu25vYnpjz5zu59hNNVdHERMnRGh-4PQ4QYoPkS_WzuivIoPcfsI2pw6_-05pjzEDRGcEsX4Mrqube-F5e4WQ_IzznzblvxPSDjKNI8vKH2IoX1rwIKWnu58A-f4uovSjd5O0QXqXf0/s320/soal+no+1+analisis+real+bartle+2.1.jpg" width="320" /></span></a></div><div style="text-align: justify;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></div><div style="text-align: justify;"><span style="font-family: "times" , "times new roman" , serif;"><i>Untuk pembahasan soal di atas, simak pembahasan di bawah ini:</i></span></div><div style="text-align: justify;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgFFoRq5I4dEadIXqWwtyux3ObnACwqhxXkSCoZ_NgIpzYV64IfMYDAozXq_nQspRQ9_xaQmygZQXzETJ5cP3K9I74gDbgtseaHS8DiGgMWHQ02x0h-KvxSieZBz8RQZ7Z_AqIIKGpGjOg/s1600/pembahasan+soal+no+1.jpg"><span style="font-family: "times" , "times new roman" , serif;"><img border="0" height="111" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgFFoRq5I4dEadIXqWwtyux3ObnACwqhxXkSCoZ_NgIpzYV64IfMYDAozXq_nQspRQ9_xaQmygZQXzETJ5cP3K9I74gDbgtseaHS8DiGgMWHQ02x0h-KvxSieZBz8RQZ7Z_AqIIKGpGjOg/s400/pembahasan+soal+no+1.jpg" width="400" /></span></a></div><div><div style="text-align: justify;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></div><div style="text-align: justify;"><span style="font-family: "times" , "times new roman" , serif;">Baca juga: </span></div></div><div><div style="text-align: justify;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></div></div><div><div style="text-align: justify;"><span style="font-family: "times" , "times new roman" , serif;"><b>Soal no 2</b></span></div></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEic1bkNsmhKsta2cuAw0dBhxGHRxfCDJkGqmnQ5TsOwZ56C8-4HLgXWRiBJ07k2BsG-Hw5qT-gabSyp_lGcR4cZZSf3nuSAE_1Dd-wtid57cZ5zuFUyFzSWvrCLNWD3DSzWgAWf5p6TaKA/s1600/soal+no+2.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><span style="font-family: "times" , "times new roman" , serif;"><img border="0" height="60" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEic1bkNsmhKsta2cuAw0dBhxGHRxfCDJkGqmnQ5TsOwZ56C8-4HLgXWRiBJ07k2BsG-Hw5qT-gabSyp_lGcR4cZZSf3nuSAE_1Dd-wtid57cZ5zuFUyFzSWvrCLNWD3DSzWgAWf5p6TaKA/s320/soal+no+2.jpg" width="320" /></span></a></div><div style="text-align: justify;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></div><div style="text-align: justify;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></div><div style="text-align: justify;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></div><div style="text-align: justify;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></div><div style="text-align: justify;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></div><div style="text-align: justify;"><span style="font-family: "times" , "times new roman" , serif;"><i>Pembahasan soal no 2</i></span></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjWcdiJxDVraNIGtz1eKWntX-tNydPB8Auz50Tp3OaggNSSm4OGrc10RdbL0HAFlsIZOtHgcjLkWR9EP-TsWmhDXMM15vwnphtGYunlDPmDhGKqsEhOpDhCOhnNVEqHNXJneY4TCOPPkGM/s1600/pembahasan+soal+no+2.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><span style="font-family: "times" , "times new roman" , serif;"><img border="0" height="101" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjWcdiJxDVraNIGtz1eKWntX-tNydPB8Auz50Tp3OaggNSSm4OGrc10RdbL0HAFlsIZOtHgcjLkWR9EP-TsWmhDXMM15vwnphtGYunlDPmDhGKqsEhOpDhCOhnNVEqHNXJneY4TCOPPkGM/s320/pembahasan+soal+no+2.jpg" width="320" /></span></a></div><div><div style="text-align: justify;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></div><div style="text-align: justify;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></div><div style="text-align: justify;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></div></div><div><div style="text-align: justify;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></div></div><div><div style="text-align: justify;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></div></div><div><div style="text-align: justify;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></div></div><div><div style="text-align: justify;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></div></div><div><div style="text-align: justify;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></div></div><div><div style="text-align: justify;"><span style="font-family: "times" , "times new roman" , serif;"><b>soal no 4,5</b></span></div></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiUfl0_GDv_whgt5sPf3Q692_mh3pehx9pY9-yvLcI5xTLDV6X3ynHhytHCwiOdb5RH91N2hRUSGYgVRwGgKcWQVeaYhSXhs9PfNmJsm-JXFNEJKcW_CPovQraZiLXjD-yuqJSYZuSS1Js/s1600/soal+4%252C5.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><span style="font-family: "times" , "times new roman" , serif;"><img border="0" height="63" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiUfl0_GDv_whgt5sPf3Q692_mh3pehx9pY9-yvLcI5xTLDV6X3ynHhytHCwiOdb5RH91N2hRUSGYgVRwGgKcWQVeaYhSXhs9PfNmJsm-JXFNEJKcW_CPovQraZiLXjD-yuqJSYZuSS1Js/s400/soal+4%252C5.jpg" width="400" /></span></a></div><div><div style="text-align: justify;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></div></div><div><div style="text-align: justify;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></div></div><div><div style="text-align: justify;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></div></div><div><div style="text-align: justify;"><br /><i><br /></i></div></div><div><div style="text-align: justify;"><span style="font-family: "times" , "times new roman" , serif;"><i>pembahasan soal no 4,5</i></span></div></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh6S0yy68tVm4H9acOtgu53NZuHKzdrWgZ2kRR94wN6trwhxp67fFwzhJr15wekTLECUWeM58Hl1M04n5DCFeEDncvezFkqXiw_AsZ4GFTY4b5gOMsLEwjNXE6fWxBpbUakKScH3SrHeXs/s1600/jawaban+4%252C5.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><span style="font-family: "times" , "times new roman" , serif;"><img border="0" height="80" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh6S0yy68tVm4H9acOtgu53NZuHKzdrWgZ2kRR94wN6trwhxp67fFwzhJr15wekTLECUWeM58Hl1M04n5DCFeEDncvezFkqXiw_AsZ4GFTY4b5gOMsLEwjNXE6fWxBpbUakKScH3SrHeXs/s400/jawaban+4%252C5.jpg" width="400" /></span></a></div><div><div style="text-align: justify;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></div></div><div><div style="text-align: justify;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></div></div><div><div style="text-align: justify;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></div></div><div><div style="text-align: justify;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></div></div><div><div style="text-align: justify;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></div></div><div><div style="text-align: justify;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></div></div><div><div style="text-align: justify;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></div></div><div><div style="text-align: justify;"><span style="font-family: "times" , "times new roman" , serif;"><b>soal no 6 </b></span></div></div><div><div style="text-align: justify;"><span style="font-family: "times" , "times new roman" , serif;">6). Buktikan bahwa tidak ada bilangan rasional s yang memenuhi s<sup>2</sup> = 6</span><br /><span style="font-family: "times" , "times new roman" , serif;"><br /></span></div></div><div class="MsoNormal"><span style="font-family: "times" , "times new roman" , serif;"><o:p></o:p></span></div><div><div style="text-align: justify;"><span style="font-family: "times" , "times new roman" , serif;"><b><i>jawab:</i></b></span><br /><div class="MsoNormal"><span style="font-family: "times" , "times new roman" , serif;">Andaikan ada bilangan rasional s yang memenuhi <i>s<sup>2</sup></i>= 6. Karena <i>s</i> adalah bilangan rasional, maka kita tuliskan <i>s = p/q</i>, untuk suatu <i>p,q </i><i>€</i><i>  Z</i> dimana<i> p</i> dan <i>q</i> relatif prima (atau dengan kata lain gcd(p,q) = 1). Sekarang,perhatikan bahwa s<sup>2</sup> = (p/q)<sup>2</sup> = p<sup>2</sup>/q<sup>2</sup>= 6 à p<sup>2</sup> = 6q<sup>2</sup>. Hal ini berarti p<sup>2</sup> adalah genap. Sebagai akibatnya p juga genap. Oleh sebab itu, maka kita bisa tuliskan p= 2m untuk suatu m € Z. Selanjutnya p<sup>2</sup></span><br /><a name='more'></a><span style="font-family: "times" , "times new roman" , serif;"> = (2m)<sup>2</sup> = 4m<sup>2</sup> = 6q<sup>2</sup> <--> 2m<sup>2</sup> = 3q<sup>2</sup>. Hal ini berarti 3q<sup>2</sup> adalah genap. Karena 3q<sup>2</sup> genap sedangkan 3 adalah ganjil, maka bisa kita simpulkan bahwa q<sup>2</sup> adalah genap. Dan sebagai </span>akibatnya, q juga genap. Namun, hal ini mengakibatkan q2 adalah genap. Dan sebagai akibatnya, q juga genap. Namun hal ini mengakibatkan bahwa p dan q sama-sama genap atau dengan kata lain p dan q tidak relative prima karena gcd(p,q) ≠ 1. Jadi, pengandaian bahwa ada bilangan rasional s yang memenuhi s2 = 6 adalah tidak benar. Dan haruslah tidak ada bilangan rasional s yang memenuhi s2 = 6.</div><div class="MsoNormal"><br /></div><div class="MsoNormal"><b>soal no 7</b></div><div class="MsoNormal">7). Buktikan bahwa tidak ada bilangan rasional t  yang memnuhi t<sup>2</sup>= 3</div><div class="MsoNormal"><br /></div><div class="MsoNormal"><o:p></o:p></div><div class="MsoNormal"><b><i>Jawab:</i></b></div><div class="MsoNormal">Andaikan ada bilangan rasional t yang memenuhi t<sup>2</sup> = 3. Karena t adalah bilangan rasional, maka kita bisa menuliskan t = a/b untuk suatu <i>a,b</i> <i style="font-family: Times, 'Times New Roman', serif;">€ Z </i>dimana <i>a</i> dan <i>b</i> relatif prima (atau dengan kata lain gcd(a,b) = 1). Sekarang perhatikan bahwa t<sup>2</sup> =  . Hal ini berarti a<sup>2</sup> habis dibagi 3. Namun hal ini mengakibatkan bahwa a juga habis dibagi 3 (mengingat jika a = 3m + 1, maka a<sup>2</sup> = (3m + 1)<sup>2</sup> = 3(3m<sup>2</sup> + 2m) + 1. Atau jika a = 3m + 2, maka a<sup>2</sup> = (3m + 2)<sup>2</sup> = 3(3m<sup>2</sup>+ 4m + 1) + 1, untuk suatu m <i style="font-family: Times, 'Times New Roman', serif;">€ Z</i> ). Selanjutnya kita bisa tuliskan a<sup>2</sup> = (3m)<sup>2 </sup>= 9m<sup>2</sup> = 3b<sup>2</sup> <--> 3m<sup>2</sup> = b<sup>2</sup>. Namun hal ini mengakibatkan b<sup>2</sup> habis dibagi 3. Dan selanjutnya, kita tahu bahwa b juga habis dibagi 3. Sehingga dapat disimpulkan bahwa a dan b sama-sama habis dibagi 3. Hal ini berkontradiksi dengan asumsi awal yang mengatakan bahwa a dan b adalah relatif prima.</div><div class="MsoNormal"><br /></div><div class="MsoNormal"><b>soal no 8</b></div><div class="MsoNormal">8). a) Tunjukkan bahwa jika x,y adalah bilangan rasional , maka x + y dan xy adalah bilangan  rasional</div><div class="MsoNormal">      b) Buktikan bahwa, jika x adalah bilangan rasional dan y adalah bilangan irrasional, maka x + y adalah bilangan irrasional. Dan jika ditambahkan syarat untuk x ≠ 0, tunjukkan bahwa xy adalah bilangan irrasional.</div><div class="MsoNormal"><br /></div><div class="MsoNormal"><b><i>jawab:</i></b></div><div class="MsoNormal"> a). Misalkan x € Q, maka x = a/b, untuk suatu <i>a,b € Z</i> dan misalkan <i>y € Q</i>, maka y = c/d, untuk suatu <i>c,d € Z</i>. Selanjutnya, perhatikan bahwa: x + y = a/b + c/d = (ad + bc)/bd <i>€ Q</i> mengingat bahwa <i>ad + cb € Z </i>dan<i> b,d € Z</i>. Kemudian dengan cara yang sama kita dapat menunjukkan bahwa x,y = a/b . c/d = ac/bd <i>€ Q</i> mengingat <i>a, c € Z </i>dan<i> b,d € Z</i>.<o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">b). Andaikan x + y € Q, maka x + y = m/n untuk suatu m,n € Z. Sekarang perhatikan bahwa: x + y = m/n <-> y = m/n – x = m/n – a/b = (mb – an)/nb. Karena <i>m b – a n </i><i>€</i><i> Z </i>dan <i>n b </i><i>€</i><i> Z</i>, maka y= (mb – an)/nb <i>€</i> <i>Q</i>yang berkontradiksi dengan hipotesis yang mengatakan y  Ɇ Q. Jadi haruslah bahwa x + y Ɇ Q. Selanjutnya andaikan x,y € Q, maka xy = p/q untuk suatu p,q € Z. Sekarang perhatikan bahwa: xy = p/q . 1/x = p/q . 1/(a/b) = p/q . b/a = pb/qa. Karena <i>p b € Z</i>, dan <i>q a € Z</i> , maka y = pb/qa € Q. Jadi, haruslah x y Ɇ Q.</div><div class="MsoNormal"><br /></div><div class="MsoNormal"><b>soal no 9. </b></div><div class="MsoNormal"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgtHMCYgZQEvA5oJKszfGmuL7aGxB9aGEwyb7yGXrMex6v6BMl-PUOwDRV5kMr6xGC1nk2_vuVHOqs5j5hNddlD9pOrvKqRYcc5ZOR7aBLm9Qv_hgdLLKWTNH9pbcft1tbOLpI3RSAqEbI/s1600/soal+no+9.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgtHMCYgZQEvA5oJKszfGmuL7aGxB9aGEwyb7yGXrMex6v6BMl-PUOwDRV5kMr6xGC1nk2_vuVHOqs5j5hNddlD9pOrvKqRYcc5ZOR7aBLm9Qv_hgdLLKWTNH9pbcft1tbOLpI3RSAqEbI/s1600/soal+no+9.jpg" /></a></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><b><i>jawab:</i></b></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEitwDyWswiVcq0tYV1KssXttwm8JtyI1U25Qbrn2kyYcj2CAXPzbU-P495O3VEzvtJhpdBfV9zpOQi_sf64dtaPALdFmbxpAgGGWIxw27ijmACOtkKIRE13vyLQVc1kfnoUH1An37n8Inw/s1600/jawaban+no+9+a.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEitwDyWswiVcq0tYV1KssXttwm8JtyI1U25Qbrn2kyYcj2CAXPzbU-P495O3VEzvtJhpdBfV9zpOQi_sf64dtaPALdFmbxpAgGGWIxw27ijmACOtkKIRE13vyLQVc1kfnoUH1An37n8Inw/s1600/jawaban+no+9+a.jpg" /></a></div><div class="MsoNormal"><b><i><br /></i></b></div><div class="MsoNormal"><b><i><br /></i></b></div><div class="MsoNormal"><b><i><br /></i></b></div><div class="MsoNormal"><b><i><br /></i></b></div><div class="MsoNormal"><b><i><br /></i></b></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjKof5ndF5ltDDTjWyDTF1Jl1NohS_lSLwKZE0JpypUyo7pyD6tG1OWUeSFoMUIjz2pJFPnYUcIRDSQQXP_C9CRRzBAzL-y5tcaXnanaBmRJL9ymUzUfzHvUQfmL3iJmqtFgVgPziJGRTA/s1600/jawaban+no+9b.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjKof5ndF5ltDDTjWyDTF1Jl1NohS_lSLwKZE0JpypUyo7pyD6tG1OWUeSFoMUIjz2pJFPnYUcIRDSQQXP_C9CRRzBAzL-y5tcaXnanaBmRJL9ymUzUfzHvUQfmL3iJmqtFgVgPziJGRTA/s1600/jawaban+no+9b.jpg" /></a></div><div class="MsoNormal"><b><i><br /></i></b></div><div class="MsoNormal"><b><i><br /></i></b></div><div class="MsoNormal"><b><i><br /></i></b></div><div class="MsoNormal"><b><i><br /></i></b></div><div class="MsoNormal"><b><i><br /></i></b></div><div class="MsoNormal"><b><i><br /></i></b></div><div class="MsoNormal"><b><i><br /></i></b></div><div class="MsoNormal"><b><i><br /></i></b></div><div class="MsoNormal"><b>soal no 10</b></div><div class="MsoNormal"><b>10.</b></div><div class="MsoNormal"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhNCWpaH5WA8nN2YlBxisW-puBMo9Rn9CJ84SuCq6nP0fd9AAQF8pZ9rv1ehY08mu0wp97aNfKWH7pw6EyEM2e_9Mrw6lKHkVKSGpYlaCmoYaNA6GVlUxfrmpRiIYsSbdFodEVYpw0wd2o/s1600/soal+no+10.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="28" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhNCWpaH5WA8nN2YlBxisW-puBMo9Rn9CJ84SuCq6nP0fd9AAQF8pZ9rv1ehY08mu0wp97aNfKWH7pw6EyEM2e_9Mrw6lKHkVKSGpYlaCmoYaNA6GVlUxfrmpRiIYsSbdFodEVYpw0wd2o/s400/soal+no+10.jpg" width="400" /></a><b><br /></b></div><div class="MsoNormal"><b><br /></b></div><div class="MsoNormal" style="text-align: left;"><br /></div><div class="MsoNormal" style="text-align: justify;"><i><b>jawab:</b></i></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEivWGv8Cp77UC_nBgcin_yAw5E-AFAKU60Y-QPDspCKuQQ-eRHiVgRIlkgKqeANZoFvDOUK-yYTTyLAh6g6MTh2L2n-VPEChzbMKRoGHTAcsZ6olKxxHzasLEWDZzMqTCkSSHBiDDtcV3Q/s1600/jawaban+no+10+a.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEivWGv8Cp77UC_nBgcin_yAw5E-AFAKU60Y-QPDspCKuQQ-eRHiVgRIlkgKqeANZoFvDOUK-yYTTyLAh6g6MTh2L2n-VPEChzbMKRoGHTAcsZ6olKxxHzasLEWDZzMqTCkSSHBiDDtcV3Q/s1600/jawaban+no+10+a.jpg" /></a></div><div class="MsoNormal"><b><br /></b></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEisHiMOg04OrHQmjv7l8kxzs3nheF4IT3_pAeByLZpth89MSfoQqh1b-65a9yZzNfBgF25JHjan977SMdoIOkbQAdlK4UGvhgDFzC2OYXNKDDpoKB7hrC6VVGUgiuhbmuIVFZ2wH4CpnaY/s1600/jawaban+no+10+b1.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEisHiMOg04OrHQmjv7l8kxzs3nheF4IT3_pAeByLZpth89MSfoQqh1b-65a9yZzNfBgF25JHjan977SMdoIOkbQAdlK4UGvhgDFzC2OYXNKDDpoKB7hrC6VVGUgiuhbmuIVFZ2wH4CpnaY/s1600/jawaban+no+10+b1.jpg" /></a></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgd4gqPF-NIcgA2UdzoX6S3nr8rgSahSNYrs7vdoafR_9tEq65Uaii4IS4NbTglb5jJ9zMLPPAsE6w5ALOSN9YC5C1Jcowjmj-sWEQxFRiApQeRyEfwqur4spOsA1vWLXqZo0o0kT5K9u8/s1600/jawaban+no+10b2.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgd4gqPF-NIcgA2UdzoX6S3nr8rgSahSNYrs7vdoafR_9tEq65Uaii4IS4NbTglb5jJ9zMLPPAsE6w5ALOSN9YC5C1Jcowjmj-sWEQxFRiApQeRyEfwqur4spOsA1vWLXqZo0o0kT5K9u8/s1600/jawaban+no+10b2.jpg" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">Setelah menyimak pembahasan analisis real 2.1 apakah kalian mengalami kesulitan? pasti tidak kan :). Nah untuk menambah pengetahuan dan kemampuan mengerjakan soal tentang analisis real, silahkan baca juga<b><span style="color: blue;"> <a href="http://funmathscience.blogspot.co.id/2015/12/lanjutan-jawaban-soal-analisis-real-1.html" target="_blank">lanjutan pembahasan soal analisis real 1. latihan 2.1 soal np 11-20</a></span></b></div><div class="MsoNormal"><b><br /></b></div><div class="MsoNormal"><b></b></div><div class="MsoNormal"><o:p></o:p></div><div class="MsoNormal"><o:p></o:p></div><div class="MsoNormal"><o:p></o:p></div><div class="MsoNormal"><o:p></o:p></div></div></div></div>

Untuk membantu anda keluar dari jerat kesulitan mengerjakan soal analisis real 1, saya akan membantu dengan postingan soal analisi real 1 yaitu pembahasan soal latihan 2.1 yang terdapat dalam buku kar...

30Nov2015

Basis dan Dimensi

November 29, 2015 Basis , bebas , bergantung , dimensi , Kombinasi , linier , matriks , Ruang , subruang , Vektor

<div dir="ltr" style="text-align: left;" trbidi="on"><div class="MsoNormal" style="line-height: 200%; text-align: justify; text-indent: 36.0pt;"><span style="font-size: large;">Pengertian basis untuk ruang vektor V serupa dengan pengertian basis untuk R<sup>n</sup>, yang telah kita kenal. Untuk mengenal basis, diperlukan pengertian membangun dan bebas linier. Pengertian membangun telah kita pelajari di materi sebelumnya yaitu <a href="http://funmathscience.blogspot.co.id/2015/08/kombinasi-linear-bebas-linear.html" target="_blank">kombinasi,bergantung, dan bebas linier</a> . Dengan pengertian bebas linier, himpunan yang membangun V dapat diperkecil sedemikian mungkin sehingga himpunan yang baru tetap membangun V.</span></div><br /><div class="MsoNormal" style="line-height: 200%; text-align: justify;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjBXM_EBU-DcWAeSW_TD7x_B8OYwbRLzSDiZy5c-qFhVHmTPgyrIFLVeEtbfREWUX2FJPVRNAiTO7U5Cm5DTOW1NlBbjlt-Bs1XQ_zimlK27bQlBq2VRx9EWiwRJa1cxq836mNcWYJqwVc/s1600/basis.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: left;"><img border="0" height="65" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjBXM_EBU-DcWAeSW_TD7x_B8OYwbRLzSDiZy5c-qFhVHmTPgyrIFLVeEtbfREWUX2FJPVRNAiTO7U5Cm5DTOW1NlBbjlt-Bs1XQ_zimlK27bQlBq2VRx9EWiwRJa1cxq836mNcWYJqwVc/s400/basis.jpg" width="400" /></a></div><span style="font-size: large;"><i><br /></i></span><span style="font-size: large;"><i><br /></i></span><span style="font-size: large;"><i></i></span><br /><a name='more'></a><span style="font-size: large;"><i><br /></i></span><span style="font-size: large;"><i>Definisi</i></span></div><div class="MsoNormal" style="line-height: 200%; text-align: justify;"><span style="font-size: large;">Contoh :</span></div><div class="MsoNormal" style="line-height: 200%; text-align: justify;"><span style="font-size: large;">Misalkan p(x) = 2 – 3x + x<sup>2</sup> , q(x) = 1 + x – x<sup>2</sup> , r(x) = 5 – 5x + x<sup>2</sup>untuk setiap x real. Karena 2p + g – r = 0 maka {p, q, r} bergantung linier di P<sub>2</sub></span></div><div class="MsoNormal" style="line-height: 200%; text-align: justify;"></div><div class="MsoNormal" style="line-height: 200%; text-align: justify;"><span style="font-size: large;"><u>Sifat</u>  :</span></div><div class="separator" style="clear: both; text-align: justify;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgnfVG04GO3r9oV6lhRfQ_z4Z513xKcCMPKEtYUbp0ZDRoxBKtSkR-M70bWsxGnt_TwHfydBjsqmTAijJv0Yaoipr9PbNl-INe4qcAnGuCrw0aLUI9-kk9R3SUasEyw3sjCWBTJvDKUYZw/s1600/basis+2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><span style="font-size: large;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgnfVG04GO3r9oV6lhRfQ_z4Z513xKcCMPKEtYUbp0ZDRoxBKtSkR-M70bWsxGnt_TwHfydBjsqmTAijJv0Yaoipr9PbNl-INe4qcAnGuCrw0aLUI9-kk9R3SUasEyw3sjCWBTJvDKUYZw/s400/basis+2.jpg" width="400" /></span></a></div><div class="separator" style="clear: both; text-align: justify;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEimrtNrITsOSsT_bFVJO2M7LHWLAl7eg8f_Bbnrck-N0dAC0rEGey6AcXd6lGtUrkjbHxeYLrKoZDl3xlsSKCOHwMHFIeH-n_61SKSjk_oz0turgWHtoffWE03ecH1XWZ4r2RejMs5mY7k/s1600/basis+3.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><span style="font-size: large;"><img border="0" height="172" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEimrtNrITsOSsT_bFVJO2M7LHWLAl7eg8f_Bbnrck-N0dAC0rEGey6AcXd6lGtUrkjbHxeYLrKoZDl3xlsSKCOHwMHFIeH-n_61SKSjk_oz0turgWHtoffWE03ecH1XWZ4r2RejMs5mY7k/s400/basis+3.jpg" width="400" /></span></a></div><div class="separator" style="clear: both; text-align: justify;"><span style="font-size: large;"><br /></span></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh1ZrQhFOqnEYwlp6tnnxL5hodx1rNWo5OcAygGv4Bi3nwzPaVm5Pq6OWVhDQtEbL4hYplE0-fRWl57ceUYlChHZm6F_kg2CVKTdaVJn0q-DicNhxLW1icRHwUZEz1ybH67dOpIJ80AG1M/s1600/basis+5.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi5hhfkbs6rpnVUqKF1ZldeV3iH-vwbIcbpvQ6fd3TaVDmdMjUjAZvONrgbJSDubTpVFt7rknLYM_9L5ScjfMID_tB3bQmJwQL-x8AdbpkjpcRnv7HfrtxRseSEqlIUwgutYiRKWthiNt4/s1600/basis+4.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em; text-align: justify;"><span style="font-size: large;"><img border="0" height="160" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi5hhfkbs6rpnVUqKF1ZldeV3iH-vwbIcbpvQ6fd3TaVDmdMjUjAZvONrgbJSDubTpVFt7rknLYM_9L5ScjfMID_tB3bQmJwQL-x8AdbpkjpcRnv7HfrtxRseSEqlIUwgutYiRKWthiNt4/s400/basis+4.jpg" width="400" /></span></a></div><div style="text-align: justify;"><span style="font-size: large;"><img border="0" height="148" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh1ZrQhFOqnEYwlp6tnnxL5hodx1rNWo5OcAygGv4Bi3nwzPaVm5Pq6OWVhDQtEbL4hYplE0-fRWl57ceUYlChHZm6F_kg2CVKTdaVJn0q-DicNhxLW1icRHwUZEz1ybH67dOpIJ80AG1M/s400/basis+5.jpg" width="400" /></span></div><br /><div style="text-align: justify;"><span style="font-size: large;"> <i>Definisi :</i></span></div><div class="MsoNormal" style="line-height: 200%; text-align: justify;"><i><span style="font-size: large;">Ruang vektor tak nol V dikatakan berdimensi hingga, jika V mempunyai basis yang hingga. Banyaknya vektor dalam suatu basis untuk V disebut dimensi (V), disingkat dim(V). dimensi ruang vektor nol didefinisikan nol.</span></i></div><div class="MsoNormal" style="line-height: 200%; text-align: justify;"><span style="font-size: large;">Contoh :</span></div><div class="MsoNormal" style="line-height: 200%; text-align: justify;"></div><ol start="1" style="margin-top: 0mm;" type="1"><li class="MsoNormal" style="line-height: 200%; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt; text-align: justify;"><span style="font-size: large;">Dimensi (<span style="font-family: "symbol"; mso-ascii-font-family: "Times New Roman"; mso-char-type: symbol; mso-hansi-font-family: "Times New Roman"; mso-symbol-font-family: Symbol;">Â</span><sup>n</sup>)      = n sebab memiliki basis yang terdiri dari n vektor.</span></li><li class="MsoNormal" style="line-height: 200%; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt; text-align: justify;"><span style="font-size: large;">Dimensi (P<sub>n</sub>) = n + 1 sebab memiliki      basis yang terdiri dari n + 1 vektor</span></li><li class="MsoNormal" style="line-height: 200%; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt; text-align: justify;"><span style="font-size: large;">Jika M<sub>2</sub> ruang vektor yang terdiri dari      matriks 2x2 dengan komponen real maka dimensi (M<sub>2</sub>) = 4, sebab M<sub>2</sub>     mempunyai basis yang terdiri dari 4 unsur.</span></li></ol><span style="font-size: large;"><u style="line-height: 200%; text-align: justify;">Sifat</u><span style="line-height: 200%; text-align: justify;"> :</span><br /><span style="line-height: 200%; text-align: justify;">     Jika V ruang vektor berdimensi n, maka :</span></span><br /><ol start="1" style="line-height: 200%; margin-top: 0mm; text-align: justify;" type="1"><li class="MsoNormal" style="line-height: 200%; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt; text-align: justify;"><span style="font-size: large;">Setiap himpunan m vektor di V dengan m > n, senantiasa      bergantung linier</span></li><li class="MsoNormal" style="line-height: 200%; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt; text-align: justify;"><span style="font-size: large;">Setiap himpunan n vektor di V yang bebas linier,      membentuk basis untuk V</span></li><li class="MsoNormal" style="line-height: 200%; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt; text-align: justify;"><span style="font-size: large;">Setiap himpunan n vektor di V yang membangun V,      membentuk basis untuk V</span></li><li class="MsoNormal" style="line-height: 200%; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt; text-align: justify;"><span style="font-size: large;">Setiap himpunan k vektor yang bebas linier di V,      dengan k < n dapat diperluas menjadi suatu basis untuk V</span></li></ol><h4 style="text-align: justify;"><span style="font-size: large;">Soal Latihan </span></h4><div class="MsoNormal" style="line-height: 200%; margin-left: 18.7pt; mso-list: l0 level1 lfo1; text-align: justify; text-indent: -18.7pt;"><span style="font-size: large;">1.<span style="font-stretch: normal; line-height: normal;">       </span>Diketahui </span></div><div class="separator" style="clear: both; text-align: justify;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgUlRZ0c36-SyCzjLjrdIlMpBZ4N-xPqcWjoxftV9y0Fm0db6zpgLkGEXxk_Pq4XS8cI7SmxkUp39Gajj_qG08ZYe8miujYrevLXEr2j8vXfJQ4a3QKsWvXybfcR3RUPMfe58xMOrUsjdk/s1600/matrixs+basis.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><span style="font-size: large;"><img border="0" height="93" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgUlRZ0c36-SyCzjLjrdIlMpBZ4N-xPqcWjoxftV9y0Fm0db6zpgLkGEXxk_Pq4XS8cI7SmxkUp39Gajj_qG08ZYe8miujYrevLXEr2j8vXfJQ4a3QKsWvXybfcR3RUPMfe58xMOrUsjdk/s200/matrixs+basis.jpg" width="200" /></span></a></div><span style="mso-text-raise: -27.0pt; position: relative; top: 27.0pt;"><span style="font-size: large;"></span></span><br /><div class="MsoNormal" style="line-height: 200%; text-align: justify;"><span style="font-size: large;">      Tunjukkan K ruang bagian dari R<sup>3</sup>. Kemudian  tentukan suatu basis untuk K</span></div><div class="MsoNormal" style="line-height: 200%; margin-left: 18.7pt; mso-list: l0 level1 lfo1; text-align: justify; text-indent: -18.7pt;"><span style="font-size: large;">2.<span style="font-stretch: normal; line-height: normal;"> </span>Tentukan suatu basis dan dimensi ruang vektor M<sub>mxn</sub></span></div><div class="MsoNormal" style="line-height: 200%; margin-left: 18.7pt; mso-list: l0 level1 lfo1; text-align: justify; text-indent: -18.7pt;"><span style="font-size: large;">3.<span style="font-stretch: normal; line-height: normal;"> </span>Tentukan suatu basis dan dimensi ruang bagian dari P<sub>n</sub>, berikut :</span></div><div class="MsoNormal" style="line-height: 200%; text-align: justify; text-indent: 18.7pt;"><span style="font-size: large;">a.  {p <span style="font-family: "symbol"; mso-ascii-font-family: "Times New Roman"; mso-char-type: symbol; mso-hansi-font-family: "Times New Roman"; mso-symbol-font-family: Symbol;">Î</span> P<sub>n</sub> | p (1) = 0}                          b.  {p <span style="font-family: "symbol"; mso-ascii-font-family: "Times New Roman"; mso-char-type: symbol; mso-hansi-font-family: "Times New Roman"; mso-symbol-font-family: Symbol;">Î</span> P<sub>n</sub> | p’(1) = 0</span>}</div></div>

Pengertian basis untuk ruang vektor V serupa dengan pengertian basis untuk Rn, yang telah kita kenal. Untuk mengenal basis, diperlukan pengertian membangun dan bebas linier. Pengertian membangun telah...

29Nov2015

Jawaban soal Kombinasi Linier, Bergantung Linier, dan Bebas Linier

November 29, 2015 bebas , bergantung , homogen , jawaban , Kombinasi , linier , persamaan , polinomial , Soal , SPL , Vektor

<div dir="ltr" style="text-align: left;" trbidi="on"><div style="text-align: justify;">Setelah kita pahami materi tentang kombinasi linier, bergantung linier dan bebas linier. sudah saatnya kita membahas jawaban dari soal di latihan pada artikel <b style="color: blue;"><a href="http://funmathscience.blogspot.co.id/2015/08/kombinasi-linear-bebas-linear.html">kombinasi linier, bergantung linier, dan bebas linier</a>. </b>Mari kita cermati pembahasan soalnya di bawah ini:</div><div style="text-align: justify;"><br /><a name='more'></a><br /></div><div style="text-align: justify;">1). <span style="background-color: white; color: #222222; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif; font-size: 13.2px; line-height: 18.48px; text-align: justify; text-indent: -17.95pt;">Misal u = (2, 4, 0), dan  v = (1,  –1, 3) adalah vektor-vektor di R</span><sup style="color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; line-height: 18.48px; text-align: justify; text-indent: -17.95pt;">3</sup><span style="background-color: white; color: #222222; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif; font-size: 13.2px; line-height: 18.48px; text-align: justify; text-indent: -17.95pt;">.</span></div><span style="background-color: white; color: #222222; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif; font-size: 13.2px; line-height: 18.48px; text-align: justify; text-indent: -17.95pt;">    </span><span style="background-color: white; color: #222222; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif; font-size: 13.2px; line-height: 18.48px; text-align: justify; text-indent: -17.95pt;">Apakah vektor berikut merupakan kombinasi linear dari vektor – vektor di atas</span><br /><span style="color: #222222; font-family: "times new roman" , serif; font-size: 10pt; line-height: 18.48px; text-align: justify; text-indent: -17.95pt;">      a.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal;">       </span></span><span style="background-color: white; color: #222222; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif; font-size: 13.2px; line-height: 18.48px; text-align: justify; text-indent: -17.95pt;">h = (4, 2, 6)</span><br /><span style="color: #222222; font-family: "times new roman" , serif; font-size: 10pt; line-height: 18.48px; text-align: justify; text-indent: -17.95pt;">      b.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal;">       </span></span><span style="background-color: white; color: #222222; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif; font-size: 13.2px; line-height: 18.48px; text-align: justify; text-indent: -17.95pt;">j = (1, 5, 6)</span><br /><span style="color: #222222; font-family: "times new roman" , serif; font-size: 10pt; line-height: 18.48px; text-align: justify; text-indent: -17.95pt;">      c.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal;">        </span></span><span style="background-color: white; color: #222222; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif; font-size: 13.2px; line-height: 18.48px; text-align: justify; text-indent: -17.95pt;">r = (0, 0, 0)</span><br /><span style="background-color: white; color: #222222; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif; font-size: 13.2px; line-height: 18.48px; text-align: justify; text-indent: -17.95pt;"><b>Jawab:</b></span><br /><div class="separator" style="clear: both; text-align: justify;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhM-cbck8APut8u8iYdjLTG2tyN37pJ-RIFRTFm2eitx2ieB8Zkfu20Zp_YsvaZ_quZqgjHbCsmZfsaR-onkMkaUjPA-yOvnY6iS2hbBr30J6TrV7fkWcW3UKxboR8ndpy-Je7PyoGVkLQ/s1600/jawaban+no+1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="jawaban kombinasi linier" border="0" height="87" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhM-cbck8APut8u8iYdjLTG2tyN37pJ-RIFRTFm2eitx2ieB8Zkfu20Zp_YsvaZ_quZqgjHbCsmZfsaR-onkMkaUjPA-yOvnY6iS2hbBr30J6TrV7fkWcW3UKxboR8ndpy-Je7PyoGVkLQ/s320/jawaban+no+1.jpg" title="jawaban no 1 kombinasi linier" width="320" /></a></div><div class="separator" style="clear: both; text-align: justify;">           ini juga dapat ditulis menjadi</div><div class="separator" style="clear: both; text-align: justify;">          </div><div class="separator" style="clear: both; text-align: justify;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi2SWfnq-iYS-4okeBl-sRvSb3ckgGl7DwcvkVfg4Hfr_rD5I22diSeCL7KgA0QaAgthvpFgA7UKCjEo4EuPLxVCJz7ltuFIRMrmRv-KMD6eikRPU2w8774EgUfQ0fIvGYJDGwvAsRKS4c/s1600/lanjutan+jawaban+no+1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="lanjutan soal kombinasi no 1" border="0" height="166" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi2SWfnq-iYS-4okeBl-sRvSb3ckgGl7DwcvkVfg4Hfr_rD5I22diSeCL7KgA0QaAgthvpFgA7UKCjEo4EuPLxVCJz7ltuFIRMrmRv-KMD6eikRPU2w8774EgUfQ0fIvGYJDGwvAsRKS4c/s320/lanjutan+jawaban+no+1.jpg" title="lanjutan soal no 1" width="320" /></a></div><div class="separator" style="clear: both; text-align: justify;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_2L9dz1OHTYYy2DHO8p5k5WOhF_ep3PmYKg_leFXMLE9TqrxB8ff71YcIvbyI9Io0Zay8huIscQtcS2iFDQrlYvZHjjzeRlIv8yNpSBVdkArxDkHISkzrrgwdJJIJQ20sbW3YyG_IHic/s1600/jawaban+no+1b.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="jawaban kombinasi linier no 1b" border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_2L9dz1OHTYYy2DHO8p5k5WOhF_ep3PmYKg_leFXMLE9TqrxB8ff71YcIvbyI9Io0Zay8huIscQtcS2iFDQrlYvZHjjzeRlIv8yNpSBVdkArxDkHISkzrrgwdJJIJQ20sbW3YyG_IHic/s320/jawaban+no+1b.jpg" title="jawaban no 1b" width="317" /></a></div><div class="separator" style="clear: both; text-align: justify;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh3pscGSAP8HcTuPjLma6Kv4S8A4vLzWgrShfKhRYihAe-IftAxwkTgkz02DKxiOA_4XqeD-xnCemiC0CfsFHkAjG4Ievj_51IPfgxDHHPVF0wBJORk4BmUYBhKgH6ZjQD0UGxcpxuT4hY/s1600/jawaban+no+1c.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="jawaban kombinasi linier 1c" border="0" height="29" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh3pscGSAP8HcTuPjLma6Kv4S8A4vLzWgrShfKhRYihAe-IftAxwkTgkz02DKxiOA_4XqeD-xnCemiC0CfsFHkAjG4Ievj_51IPfgxDHHPVF0wBJORk4BmUYBhKgH6ZjQD0UGxcpxuT4hY/s320/jawaban+no+1c.jpg" title="jawaban no 1c" width="320" /></a></div><div class="separator" style="clear: both; text-align: justify;">          </div><div class="separator" style="clear: both; text-align: justify;">    <a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh9Mv6OWk-XkQTG6iSf6OD2aHL-dNhyX_a2wQWmpjGtfpqumfCK0H5JDARHl6yN1adfkWUxSl1XnLjvlE9sUk8_KlCXL1FrQNNQ2VUGhyphenhyphenYj6D-6DL0aEH_ir_b8MFyQGYGI7GUjGYvf_8Y/s1600/jawaban+lanjutan+no+1c.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="lanjutan jawaban kombinasi linier no 1c" border="0" height="64" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh9Mv6OWk-XkQTG6iSf6OD2aHL-dNhyX_a2wQWmpjGtfpqumfCK0H5JDARHl6yN1adfkWUxSl1XnLjvlE9sUk8_KlCXL1FrQNNQ2VUGhyphenhyphenYj6D-6DL0aEH_ir_b8MFyQGYGI7GUjGYvf_8Y/s320/jawaban+lanjutan+no+1c.jpg" title="lanjutan jawaban no 1c" width="320" /></a></div><div class="separator" style="clear: both; text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: justify;">2).  <span style="background-color: white; color: #222222; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif; font-size: 13.2px; line-height: 18.48px; text-align: justify; text-indent: -23.9333px;">Diketahui v =(3,9,-4,-2) merupakan kombinasi linier u</span><sub style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; text-align: justify; text-indent: -23.9333px;">1</sub><span style="background-color: white; color: #222222; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif; font-size: 13.2px; line-height: 18.48px; text-align: justify; text-indent: -23.9333px;">= (1,-2,0,3), u</span><sub style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; text-align: justify; text-indent: -23.9333px;">2</sub><span style="background-color: white; color: #222222; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif; font-size: 13.2px; line-height: 18.48px; text-align: justify; text-indent: -23.9333px;"> = (2,3,0,-1) dan u</span><sub style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; text-align: justify; text-indent: -23.9333px;">3</sub><span style="background-color: white; color: #222222; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif; font-size: 13.2px; line-height: 18.48px; text-align: justify; text-indent: -23.9333px;">= (2,-1,2,1). Apakah v merupakan kombinasi linear dari u</span><sub style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; text-align: justify; text-indent: -23.9333px;">1</sub><span style="background-color: white; color: #222222; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif; font-size: 13.2px; line-height: 18.48px; text-align: justify; text-indent: -23.9333px;"> , u</span><sub style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; text-align: justify; text-indent: -23.9333px;">2</sub><span style="background-color: white; color: #222222; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif; font-size: 13.2px; line-height: 18.48px; text-align: justify; text-indent: -23.9333px;">dan u</span><sub style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; text-align: justify; text-indent: -23.9333px;">3</sub><span style="background-color: white; color: #222222; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif; font-size: 13.2px; line-height: 18.48px; text-align: justify; text-indent: -23.9333px;"> ?</span></div><div class="separator" style="clear: both; text-align: justify;"><span style="background-color: white; color: #222222; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif; font-size: 13.2px; line-height: 18.48px; text-align: justify; text-indent: -23.9333px;">Jawab: </span></div><div class="separator" style="clear: both; text-align: justify;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhArz7JngehKmUHWlNCBuYM_eI_XH2CdDyrNLAvfOZfyW3LQ2-2iTMzUTFzU1NlR0FBQ9zPBDvamW4BupKdxsrdL0PbRj8AN_Wa9rGgbELyhNXXXPk1jAfw-WR3YkhbSJFkk0vzwyXHxMo/s1600/jawaban+no+2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: left;"><img border="0" height="123" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhArz7JngehKmUHWlNCBuYM_eI_XH2CdDyrNLAvfOZfyW3LQ2-2iTMzUTFzU1NlR0FBQ9zPBDvamW4BupKdxsrdL0PbRj8AN_Wa9rGgbELyhNXXXPk1jAfw-WR3YkhbSJFkk0vzwyXHxMo/s320/jawaban+no+2.jpg" width="320" /></a></div><div class="separator" style="clear: both; text-align: justify;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgjybHakYboB_h0i-wPqVGrNArDHHUAF0O_15bFiM2qJ3M1dXY4NUaq_uWVlNbVIHFGRQMP1C1vx9sjC539G9RegiUcKG-2HPnghZhkfQ83dUuc29G7o2Zz2yIj2gknoinCnBp0Hwtu9uk/s1600/lanjutan+jawaban+no+2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="162" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgjybHakYboB_h0i-wPqVGrNArDHHUAF0O_15bFiM2qJ3M1dXY4NUaq_uWVlNbVIHFGRQMP1C1vx9sjC539G9RegiUcKG-2HPnghZhkfQ83dUuc29G7o2Zz2yIj2gknoinCnBp0Hwtu9uk/s320/lanjutan+jawaban+no+2.jpg" width="320" /></a></div><div class="separator" style="clear: both; text-align: justify;">    Penyelesaiannya: x = 1, y = 3, dan z = -2</div><div class="separator" style="clear: both; text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: justify;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgGaKsGyCaIyUl0PZuYEH8ji4q6y3U46lApqGkjKWLBJ1idGerwARAXe1c7UCLK__sg5p3EtKYltfzSsvF5Pe4IdnXiT2kVv6oIwSOdpYCZ8IrWLcYO1M_nvzA1vLoQOmwImFCs8XYFLH0/s1600/lanjutan+soal+no+2.1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="88" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgGaKsGyCaIyUl0PZuYEH8ji4q6y3U46lApqGkjKWLBJ1idGerwARAXe1c7UCLK__sg5p3EtKYltfzSsvF5Pe4IdnXiT2kVv6oIwSOdpYCZ8IrWLcYO1M_nvzA1vLoQOmwImFCs8XYFLH0/s200/lanjutan+soal+no+2.1.jpg" width="200" /></a></div><div class="separator" style="clear: both; text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: justify;">3). <span style="background-color: white; color: #222222; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif; font-size: 13.2px; line-height: 18.48px; text-indent: -23.9333px;">Apakah polinomial-polinomial berikut ini bebas linier ?       </span></div><div class="separator" style="clear: both; text-align: justify;"><span style="background-color: white; color: #222222; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif; font-size: 13.2px; line-height: 18.48px; text-indent: -23.9333px;"><b>p</b></span><sub style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; text-indent: -23.9333px;"><b>1</b></sub><span style="background-color: white; color: #222222; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif; font-size: 13.2px; line-height: 18.48px; text-indent: -23.9333px;"> = 1 – 2x + 3 x</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; text-indent: -23.9333px;">2</sup><span style="background-color: white; color: #222222; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif; font-size: 13.2px; line-height: 18.48px; text-indent: -23.9333px;"> , </span></div><div class="separator" style="clear: both; text-align: justify;"><span style="background-color: white; color: #222222; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif; font-size: 13.2px; line-height: 18.48px; text-indent: -23.9333px;"><b>p</b></span><sub style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; text-indent: -23.9333px;"><b>2</b></sub><span style="background-color: white; color: #222222; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif; font-size: 13.2px; line-height: 18.48px; text-indent: -23.9333px;"> = 5 + 6x – x</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; text-indent: -23.9333px;">2</sup><span style="background-color: white; color: #222222; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif; font-size: 13.2px; line-height: 18.48px; text-indent: -23.9333px;"> , </span></div><div class="separator" style="clear: both; text-align: justify;"><b><span style="background-color: white; color: #222222; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif; font-size: 13.2px; line-height: 18.48px; text-indent: -23.9333px;">p</span><sub style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; text-indent: -23.9333px;">3</sub></b><span style="background-color: white; color: #222222; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif; font-size: 13.2px; line-height: 18.48px; text-indent: -23.9333px;"> = 3 + 2x + x</span><sup style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; text-indent: -23.9333px;">2</sup></div><div class="separator" style="clear: both; text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: justify;"><b>Jawab:</b></div><div class="separator" style="clear: both; text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: justify;">Untuk menguji polinomial bebas atau bergantung linier, langkah yang dilakukan adalah dengan menuliskan persamaan homogen sebagai berikut:</div><div class="separator" style="clear: both; text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: left;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj4idUk3KPKwyVIYO1HyP9Bs8hsKyU701VTo51kdV4AKrCBZk71FVd-KcHAaVNFz84tl8rwTZV8QBeuYDH0AQ45xynqtKspR257Ef_2S1pwXdiykGHf-_huleZl_ApJQT7lXwYv5p5RTLw/s1600/jawaban+no+3.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="83" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj4idUk3KPKwyVIYO1HyP9Bs8hsKyU701VTo51kdV4AKrCBZk71FVd-KcHAaVNFz84tl8rwTZV8QBeuYDH0AQ45xynqtKspR257Ef_2S1pwXdiykGHf-_huleZl_ApJQT7lXwYv5p5RTLw/s320/jawaban+no+3.jpg" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgA7MTFp2jX0Uly4BIoVRrss5Azl6_dDbbkG283AgZWZlbmPBRN5kC2EJ9Sx_DxedDQeQNA5TgBuzVoyz9mSQk0RSvlBH0-LVqp8BY5AHoCW6Osz2Shn2v3wHngC9CwHvd4BvqiqDEY0pM/s1600/jawaban+no+3.2.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="167" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgA7MTFp2jX0Uly4BIoVRrss5Azl6_dDbbkG283AgZWZlbmPBRN5kC2EJ9Sx_DxedDQeQNA5TgBuzVoyz9mSQk0RSvlBH0-LVqp8BY5AHoCW6Osz2Shn2v3wHngC9CwHvd4BvqiqDEY0pM/s320/jawaban+no+3.2.jpg" width="320" /></a></div><div class="separator" style="clear: both; text-align: left;">4.) <span style="background-color: white; color: #222222; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif; font-size: 13.2px; line-height: 18.48px; text-align: justify; text-indent: -17.95pt;">Diketahui u=(1,2), v=(2,2),  w=(1,3). Tentukan:                       </span></div><div class="normal" style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px; line-height: 18.48px; margin-left: 42.55pt; text-align: justify;"> a. Apakah  u, v  dan  w  membangun R<sup>2</sup>?</div><br style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px; line-height: 18.48px;" /><div class="normal" style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px; line-height: 18.48px; margin-left: 42.55pt; text-align: justify;">b. Apakah  u, v  dan  w  bebas linier ?</div><div class="normal" style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px; line-height: 18.48px; margin-left: 42.55pt; text-align: justify;"><b><i><u>(masih dalam penyelesaian)</u></i></b></div><div class="normal" style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px; line-height: 18.48px; margin-left: 42.55pt; text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: left;">5). <a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj5zbKEWd-0TNvYVDwnQX83VMe3EBo5ywLNisb5oo2nVIzOscOCiVOBxBwyet27Bn5X-fBTjDrD7ZVHs8tsKyZuTBUU5gLSQjUz3lK3eVa95sariLW_IM2v8kk7dyRIPLEXL-a7dUj7ve4/s1600/soal+no+4.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="57" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj5zbKEWd-0TNvYVDwnQX83VMe3EBo5ywLNisb5oo2nVIzOscOCiVOBxBwyet27Bn5X-fBTjDrD7ZVHs8tsKyZuTBUU5gLSQjUz3lK3eVa95sariLW_IM2v8kk7dyRIPLEXL-a7dUj7ve4/s640/soal+no+4.jpg" width="640" /></a></div><div class="separator" style="clear: both; text-align: left;"><span style="background-color: white; color: #222222; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif; font-size: 13.2px; line-height: 18.48px; text-align: justify;"><b><br /></b></span></div><div class="separator" style="clear: both; text-align: left;"><span style="background-color: white; color: #222222; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif; font-size: 13.2px; line-height: 18.48px; text-align: justify;"><b>jawab:</b></span></div><div class="separator" style="clear: both; text-align: left;"><span style="background-color: white; color: #222222; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif; font-size: 13.2px; line-height: 18.48px; text-align: justify;"><br /></span></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi4MNogWe5e4lLIrePi4Vq51qAWcqsAx72OMjh9UxpvXL3252yE_ecxxQIifUA_UmKA8HPK3WRAPfrZEHGdB1BDiQ7DbzQ2Ai8TiLfgcBFuMetcIySlhS5i-rASbuHXkoMorHhE2-pN3LE/s1600/jawaban+no+4.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="195" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi4MNogWe5e4lLIrePi4Vq51qAWcqsAx72OMjh9UxpvXL3252yE_ecxxQIifUA_UmKA8HPK3WRAPfrZEHGdB1BDiQ7DbzQ2Ai8TiLfgcBFuMetcIySlhS5i-rASbuHXkoMorHhE2-pN3LE/s640/jawaban+no+4.jpg" width="640" /></a></div><div class="MsoNormal" style="line-height: 150%; text-align: justify; text-justify: inter-ideograph;"><span lang="IN">Penyamaan komponen-komponen yang bersesuaian memberikan:</span></div><div class="MsoNormal" style="line-height: 150%; text-align: justify; text-justify: inter-ideograph;"><i><span lang="IN">x</span></i><span lang="IN"> – 2<i>y</i> = 6</span></div><div class="MsoNormal" style="line-height: 150%; text-align: justify; text-justify: inter-ideograph;"><span lang="IN">2<i>x</i> + <i>y</i> = 2.</span></div><div class="MsoNormal" style="line-height: 150%; text-align: justify; text-justify: inter-ideograph;"><span lang="IN"></span></div><div class="MsoNormal" style="line-height: 150%; text-align: justify; text-justify: inter-ideograph;"><span lang="IN">Bentuk ini adalah sistem persamaan linier yang terdiri dari dua persamaan dan dua bilangan yang tidak diketahui. Untuk memecahkan sistem persamaan linier ini dapat digunakan eliminasi Gauss-Jordan pada matriks yang dipernesarnya, yaitu:</span><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjiWqSqvrW33Sez8YUWLFD0hLanX5xydKqjyEcBOF-Rb21AZ9kRkv2f4ESMFqbistDG_Es3T3ROLzD3rruVtHLZO8j1HBl1a-cVSdvvqHFLPrlpxB9nEL62JID1TSstSOoO_s-0XBuyiBk/s1600/jawaban+no+4.2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="136" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjiWqSqvrW33Sez8YUWLFD0hLanX5xydKqjyEcBOF-Rb21AZ9kRkv2f4ESMFqbistDG_Es3T3ROLzD3rruVtHLZO8j1HBl1a-cVSdvvqHFLPrlpxB9nEL62JID1TSstSOoO_s-0XBuyiBk/s640/jawaban+no+4.2.jpg" width="640" /></a></div><div class="MsoNormal" style="line-height: 150%; text-align: justify; text-justify: inter-ideograph;"><span lang="IN"><br /></span></div><div class="MsoNormal" style="line-height: 150%; text-align: justify; text-justify: inter-ideograph;"><span lang="IN">Setelah mencermati Jawaban Latihan Soal Kombinasi Linier, Bergantung Linier, dan Bebas Linier. pasti kalian sudah bisa memahami lebih jauh tentang bab vektor. Terima kasih telah mengunjungi blog ini, semoga bermanfaat. </span></div><div class="separator" style="clear: both; text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: justify;"><br /></div></div>

Setelah kita pahami materi tentang kombinasi linier, bergantung linier dan bebas linier. sudah saatnya kita membahas jawaban dari soal di latihan pada artikel kombinasi linier, bergantung linier, dan ...

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