{"id":71,"date":"2023-09-16T13:04:23","date_gmt":"2023-09-16T13:04:23","guid":{"rendered":"https:\/\/mathority.org\/id\/kombinasi-linier-vektor-contoh-latihan-yang-diselesaikan\/"},"modified":"2023-09-16T13:04:23","modified_gmt":"2023-09-16T13:04:23","slug":"kombinasi-linier-vektor-contoh-latihan-yang-diselesaikan","status":"publish","type":"post","link":"https:\/\/mathority.org\/id\/kombinasi-linier-vektor-contoh-latihan-yang-diselesaikan\/","title":{"rendered":"Kombinasi vektor linier"},"content":{"rendered":"<p>Pada halaman ini Anda akan menemukan penjelasan tentang apa yang dimaksud dengan kombinasi linier antar vektor. Selain itu, Anda akan dapat melihat contoh bagaimana vektor dinyatakan sebagai kombinasi linier dan, sebagai tambahan, Anda akan dapat berlatih dengan latihan dan soal yang diselesaikan langkah demi langkah. <\/p>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-104\"><\/div>\n<\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfque-es-la-combinacion-lineal-de-vectores\"><\/span> Apa yang dimaksud dengan kombinasi linier vektor?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Pengertian kombinasi linier adalah sebagai berikut: <\/p>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-105\"><\/div>\n<\/div>\n<div style=\"background-color:#FFCC8080;padding-top: 20px; padding-bottom: 0.5px; padding-right: 40px; padding-left: 30px; border: 2px solid #FFB74D; border-radius:20px;\">\n<p style=\"text-align:left\"> <strong>Kombinasi linier<\/strong> suatu himpunan vektor adalah vektor yang diperoleh dengan menjumlahkan seluruh vektor pada himpunan tersebut dikalikan skalar (bilangan real).<\/p>\n<p style=\"text-align:left\"> Dengan kata lain, diberikan himpunan vektor<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-33729e6d20b00643b5d9ddf38544c11c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{v}}_1, \\vv{\\text{v}}_2,\\ldots \\vv{\\text{v}}_n,\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"96\" style=\"vertical-align: -4px;\"><\/p>\n<p> kombinasi liniernya adalah:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a1fe2e85f82aa1452aa43a172ca8d256_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{w}}=a_1\\vv{\\text{v}}_1+a_2\\vv{\\text{v}}_2+\\dots + a_n\\vv{\\text{v}}_n\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"226\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p style=\"text-align:left\"> Dimana koefisiennya<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f91083f3035e5168a6f0b3e6335d6858_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a_i\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"14\" style=\"vertical-align: -3px;\"><\/p>\n<p> Ini adalah bilangan real.<\/p>\n<\/div>\n<p> Oleh karena itu, vektor yang merupakan kombinasi linier dari vektor-vektor lain berarti vektor pertama dapat dinyatakan dalam vektor kedua.<\/p>\n<p> Konsep ini dapat lebih dipahami dengan membuat grafik sebuah vektor pada bidang yang merupakan kombinasi linier dari dua vektor: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/combinaison-lineaire-de-vecteurs-graphique.webp\" alt=\"kombinasi linier vektor di r3\" class=\"wp-image-781\" width=\"405\" height=\"408\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Seperti yang Anda lihat pada representasi grafis di atas, vektor<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3b4bbbc56786695092eac40831aee80d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{w}}\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"13\" style=\"vertical-align: 0px;\"><\/p>\n<p> dapat diperoleh dari vektor<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cac24ae79c1e4cbc459f01ed5e4f824e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}}\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> Dan<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-391ac2e3ba0b7f327ba5a0edc1ba162d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{v}}\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> melakukan operasi vektor. Oleh karena itu, vektor<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3b4bbbc56786695092eac40831aee80d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{w}}\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"13\" style=\"vertical-align: 0px;\"><\/p>\n<p> adalah kombinasi linier dari dua vektor lainnya.<\/p>\n<p> Perlu ditekankan bahwa kombinasi linier ini bersifat <strong>unik<\/strong> , atau dengan kata lain hanya ada satu kombinasi linier yang layak untuk setiap vektor. Karena mengikuti contoh sebelumnya, jika kita mengalikannya<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cac24ae79c1e4cbc459f01ed5e4f824e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}}\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> untuk 6, bukan 4, kita akan memperoleh vektor lain yang berbeda. <\/p>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-106\"><\/div>\n<\/div>\n<p> Selain itu, salah satu sifat dari kombinasi linier pada bidang (dalam R2) adalah bahwa suatu vektor dapat dikatakan sebagai kombinasi linier dari dua vektor lainnya jika keduanya mempunyai arah yang berbeda, yaitu jika keduanya tidak sejajar.<\/p>\n<p> Selain itu, terkadang kita dapat mengidentifikasi dengan jelas bahwa dua vektor merupakan kombinasi linier. Untuk melakukan ini, komponen-komponennya cukup <strong>proporsional<\/strong> . Misalnya, koordinat dua vektor berikut adalah proporsional, sehingga vektor-vektor tersebut merupakan kombinasi linier:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f7e90b69f6225543322e762773bbe775_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} = (1,2,-1) \\qquad \\vv{\\text{v}} = (3,6,-3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"241\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-aac41542948764e158ebe590c6b36e67_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{3}{1} = \\cfrac{6}{2} = \\cfrac{-3}{-1} = 3 \\ \\longrightarrow \\ \\text{Proporcionales}\\ \\longrightarrow \\ \\begin{array}{c} \\text{Combinaci\\'on}\\\\[2ex] \\text{lineal} \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"51\" width=\"456\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Terakhir, baik dalam ruang vektor dua dimensi (dalam R2) atau tiga dimensi (dalam R3), jika terdapat kombinasi linier dalam sekumpulan vektor, hal ini menunjukkan bahwa vektor-vektor tersebut saling <strong>bergantung secara linier<\/strong> . Sebaliknya, jika tidak ada kombinasi linier yang mungkin terjadi antar vektor, maka vektor-vektor tersebut <strong>bebas linier<\/strong> .<\/p>\n<p> Jika konsep terakhir ini belum sepenuhnya jelas bagi Anda, kami sarankan untuk membaca penjelasan kami tentang <a href=\"https:\/\/mathority.org\/id\/vektor-bebas-dan-bergantung-linier-kemandirian-ketergantungan-linier\/\">vektor bergantung linier dan bebas<\/a> , di sini Anda akan menemukan apa yang dimaksud dengan vektor bergantung atau bebas linier, contoh masing-masing jenis dan perbedaan di antara keduanya. . Konsep ini banyak digunakan dan bahkan banyak ditanyakan dalam ujian, jadi penting bagi Anda untuk memahaminya dengan baik. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"como-expresar-un-vector-como-combinacion-lineal-de-otros-vectores\"><\/span> Bagaimana menyatakan suatu vektor sebagai kombinasi linier dari vektor-vektor lainnya <span class=\"ez-toc-section-end\"><\/span><\/h2>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-109\"><\/div>\n<\/div>\n<p> Kita kemudian akan melihat bagaimana menyelesaikan masalah umum di mana kita diminta untuk menemukan kombinasi linier suatu vektor.<\/p>\n<ul>\n<li> Ekspresikan vektornya\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-88e41d561c3898029b7b94d7014c1e79_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{x}}\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> sebagai kombinasi linier dari<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4b6be5a59bbf478047e4f3ace338ee48_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}}, \\vv{\\text{v}}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: -4px;\"><\/p>\n<p> Dan <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d8af8ced46d93e73dc5290e0cca4dc6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{w}}.\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"17\" style=\"vertical-align: 0px;\"><\/p>\n<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7c6a832874f83ba4de52e88fdd6ed48a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{x}} = (3,1,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"88\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-746bff339baec38ef705a9ede42411cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} = (1,0,1) \\qquad \\vv{\\text{v}} = (1,2,0) \\qquad \\vv{\\text{w}} = (0,1,-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"355\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Sehingga vektornya<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-88e41d561c3898029b7b94d7014c1e79_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{x}}\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> merupakan kombinasi linier dari vektor-vektor lainnya, maka persamaan berikut harus dipenuhi:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-06d3d6ec5ca4921b109f8f974e73cbbd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a_1\\vv{\\text{u}}+a_2\\vv{\\text{v}}+ a_3\\vv{\\text{w}}=\\vv{\\text{x}}\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"160\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p> Dimana koefisiennya<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-41a350e61a3992febcf5f69fdb79f79a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a_1, a_2\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"41\" style=\"vertical-align: -4px;\"><\/p>\n<p> Dan<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a4306749a1a62a769b17b849d10edba8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a_3\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"16\" style=\"vertical-align: -3px;\"><\/p>\n<p> Inilah hal-hal yang tidak diketahui yang harus kita temukan.<\/p>\n<p> Oleh karena itu kami mengganti setiap vektor dengan koordinatnya:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d9ed95a00184b48d358ba1b0a2abf105_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a_1\\begin{pmatrix} 1 \\\\0\\\\1 \\end{pmatrix}+a_2\\begin{pmatrix} 1 \\\\2\\\\0 \\end{pmatrix}+ a_3\\begin{pmatrix} 0 \\\\1\\\\-1 \\end{pmatrix}=\\begin{pmatrix} 3 \\\\1\\\\2 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"296\" style=\"vertical-align: -27px;\"><\/p>\n<\/p>\n<p> Kami mengalikan setiap vektor dengan koefisiennya:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-626790fc18c5942db14924be2397c9f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix} a_1 \\\\0\\\\a_1 \\end{pmatrix}+\\begin{pmatrix} a_2 \\\\2a_2\\\\0 \\end{pmatrix}+ \\begin{pmatrix} 0 \\\\a_3\\\\-a_3 \\end{pmatrix}=\\begin{pmatrix} 3 \\\\1\\\\2 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"264\" style=\"vertical-align: -27px;\"><\/p>\n<\/p>\n<p> Kami menambahkan vektor:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1f8ab5661ba692df579d8e88b6244cdd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix} a_1 +a_2\\\\2a_2+a_3\\\\a_1-a_3 \\end{pmatrix}=\\begin{pmatrix} 3 \\\\1\\\\2 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"150\" style=\"vertical-align: -27px;\"><\/p>\n<\/p>\n<p> Setiap koordinat kiri harus sama dengan setiap koordinat kanan. Oleh karena itu kami memiliki 3 persamaan:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8e5fe050102a285a325dcd81d07ef5d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left. \\begin{array}{l} a_1 +a_2 = 3 \\\\[2ex] 2a_2+a_3 =1\\\\[2ex] a_1-a_3 = 2 \\end{array} \\right\\}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"118\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Yang tersisa hanyalah menyelesaikan sistem persamaan yang diperoleh. Untuk melakukannya, gunakan metode yang Anda sukai (metode substitusi, aturan Cramer, metode Gauss-Jordan, dll), dalam hal ini kita akan menggunakan metode Gauss: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8aa4e245614f286e0697797a18ba4465_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left( \\begin{array}{ccc|c} 1&amp;1&amp;0&amp; 3 \\\\[2ex] 0&amp;2&amp;1&amp;1\\\\[2ex] 1&amp;0&amp;-1&amp;2 \\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"135\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-41f1d9c941fe239bb40297b998eb6929_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left(\\begin{array}{ccc|c} 1&amp;1&amp;0&amp; 3 \\\\[2ex] 0&amp;2&amp;1&amp;1\\\\[2ex] 1&amp;0&amp;-1&amp;2 \\end{array} \\right) \\begin{array}{c} \\\\[2ex] \\\\[2ex] \\xrightarrow{F_3-F_1} \\end{array} \\left( \\begin{array}{ccc|c} 1&amp;1&amp;0&amp; 3 \\\\[2ex] 0&amp;2&amp;1&amp;1\\\\[2ex] 0&amp;-1&amp;-1&amp;-1 \\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"382\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-02a8a00406479f367627b682099e05c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left( \\begin{array}{ccc|c} 1&amp;1&amp;0&amp; 3 \\\\[2ex] 0&amp;2&amp;1&amp;1\\\\[2ex] 0&amp;-1&amp;-1&amp;-1 \\end{array} \\right)\\begin{array}{c} \\\\[2ex] \\\\[2ex] \\xrightarrow{2F_3+F_2}\\end{array} \\left( \\begin{array}{ccc|c} 1&amp;1&amp;0&amp; 3 \\\\[2ex] 0&amp;2&amp;1&amp;1\\\\[2ex] 0&amp;0&amp;-1&amp;-1 \\end{array}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"403\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Oleh karena itu, sistem langkah-langkah yang diperoleh adalah:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-74ed1b18779582d6683ecaa1a9085e3d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left. \\begin{array}{r} a_1 +a_2 = 3 \\\\[2ex] 2a_2+a_3 =1\\\\[2ex] -a_3 = -1 \\end{array} \\right\\}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"118\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Yang harus kita lakukan sekarang adalah mengklarifikasi hal-hal yang tidak diketahui dan menemukan nilainya. Jadi dari persamaan terakhir kita temukan<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a5e5ed86162a9b0324b8f44dc16fcbce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a_3:\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"26\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a9098f1754f21ebdb169710a81771238_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-a_3 = -1 \\ \\longrightarrow \\ \\bm{a_3 = 1}\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"175\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p> Dari persamaan kedua sistem, kita menghitung nilai <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2789190f1df15f5bd570b643d9bb29f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a_2:\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"26\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5d375653cd224859cfb1172eff34b13a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2a_2+a_3 =1 \\ \\xrightarrow{a_3\\ = \\ 1} \\ 2a_2+1=1\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"261\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cd6833a5f5007dec00e1b7a1c0820bd9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2a_2=1-1\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"88\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-aa265a6ea06995349079b84bfae9d627_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2a_2=0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"59\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0d26904a10ba1c4d37589b41962c6b9b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{a_2=0}\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"50\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p> Dan akhirnya, dari persamaan pertama sistem langkah, kita menemukan hal yang tidak diketahui<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c80696de686104689a20cb70c0033830_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a_1:\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"26\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9506e180ee4e8b7a69fa509b823fdcca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a_1 +a_2 = 3 \\ \\xrightarrow{a_3\\ = \\ 1 \\ ; \\ a_2 \\ = \\ 0 } \\ \\bm{a_1 = 3}\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"273\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p> Oleh karena itu, penyelesaian sistem persamaan linear adalah:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f27368cbdc2111d5e30c1c29c5da8f95_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a_1=3 \\qquad a_2=0 \\qquad a_3 = 1\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"219\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p> Jadi vektornya<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-88e41d561c3898029b7b94d7014c1e79_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{x}}\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> Hal ini dapat dinyatakan dengan kombinasi linier berikut: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-80aba06b670bf9eedd4074be0750c3d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{x}}= a_1\\vv{\\text{u}}+a_2\\vv{\\text{v}}+ a_3\\vv{\\text{w}}\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"160\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0cad8a3d5bdbe0461d347a8a3f21f794_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{x}}= 3\\vv{\\text{u}}+0\\vv{\\text{v}}+ 1\\vv{\\text{w}}\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"135\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8ccdc9d2a3852c38c4442d0b601b6644_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\mathbf{x}}\\bm{= 3}\\vv{\\mathbf{u}} \\bm{+} \\vv{\\mathbf{w}}}\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"78\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p> Oleh karena itu, secara efektif terdapat ketergantungan linier antar vektor. Di sisi lain, jika tidak ada solusi sistem persamaan yang diperoleh, ini berarti vektor<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-88e41d561c3898029b7b94d7014c1e79_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{x}}\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> Ia bebas linier terhadap vektor-vektor lain dan, oleh karena itu, tidak ada kombinasi linier yang mungkin untuk memperoleh vektor tersebut dari vektor-vektor lain. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-combinacion-lineal-de-vectores\"><\/span> Latihan soal kombinasi linier vektor<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> Latihan 1<\/h3>\n<p> Di antara tiga vektor berikut, tunjukkan pasangan mana yang merupakan kombinasi linier satu sama lain. Selain itu, carilah hubungan kombinasi linier dari pasangan vektor tersebut. <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0558431e1c2e3040ed06e8bd04be0d3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} = (2,4,3) \\qquad \\vv{\\text{v}} = (1,2,-3) \\qquad \\vv{\\text{w}} = (-3,-6,9)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"383\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E4F0FE\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E4F0FE\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Lihat solusinya<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Untuk mengetahui apakah suatu pasangan vektor merupakan kombinasi linier, kita harus melihat apakah koordinatnya sebanding.<\/p>\n<p class=\"has-text-align-left\"> Kita cek dulu vektornya<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cac24ae79c1e4cbc459f01ed5e4f824e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}}\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> dengan vektornya<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8f5713006a9840d2d71efbe7b540d21a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{v}} :\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"18\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fc4cadf576dfcd515bba9e31c113c317_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{2}{1} = \\cfrac{4}{2} \\neq \\cfrac{3}{-3} \\ \\longrightarrow \\ \\text{No proporcionales}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"283\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kedua, kita periksa vektornya<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cac24ae79c1e4cbc459f01ed5e4f824e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}}\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> dengan vektornya<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-97cea7925862c08ac4cf5b4963c0187b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{w}} :\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"22\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ccc5afad1474f92824813625a0f04242_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{2}{-3} = \\cfrac{4}{-6} \\neq \\cfrac{3}{9} \\ \\longrightarrow \\ \\text{No proporcionales}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"297\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Terakhir, kami menguji vektornya<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-391ac2e3ba0b7f327ba5a0edc1ba162d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{v}}\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> dengan vektornya<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-97cea7925862c08ac4cf5b4963c0187b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{w}} :\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"22\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f818eb5ae0825dd43290331519599c21_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{1}{-3} = \\cfrac{2}{-6} = \\cfrac{-3}{9} = -\\cfrac{1}{3} \\ \\longrightarrow \\ \\text{Proporcionales}\\ \\longrightarrow \\ \\begin{array}{c} \\text{Combinaci\\'on}\\\\[2ex] \\text{lineal} \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"51\" width=\"499\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Jadi satu-satunya pasangan vektor yang merupakan kombinasi linier adalah<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-391ac2e3ba0b7f327ba5a0edc1ba162d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{v}}\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> Dan<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d8af8ced46d93e73dc5290e0cca4dc6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{w}}.\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"17\" style=\"vertical-align: 0px;\"><\/p>\n<p> Selanjutnya hubungan mereka adalah sebagai berikut:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ca9417b2ef9db0db6d78c0af39dde0b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{v}}= -\\cfrac{1}{3} \\vv{\\text{w}}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"71\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Atau setara:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-69433589474e50574aa5d9dcbd188b28_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{w}}= -3 \\vv{\\text{v}}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"68\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Meskipun pernyataan tersebut tidak mengharuskannya, satu-satunya vektor yang bergantung secara linear satu sama lain adalah<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-391ac2e3ba0b7f327ba5a0edc1ba162d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{v}}\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> Dan<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3b4bbbc56786695092eac40831aee80d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{w}}\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"13\" style=\"vertical-align: 0px;\"><\/p>\n<p> karena terdapat kombinasi linear diantara keduanya. Pasangan-pasangan lainnya bebas linier, karena tidak dapat digabungkan secara linier.<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Latihan 2<\/h3>\n<p> Temukan hubungan linier antara vektor<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-88e41d561c3898029b7b94d7014c1e79_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{x}}\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> dan himpunan vektor<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4b6be5a59bbf478047e4f3ace338ee48_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}}, \\vv{\\text{v}}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: -4px;\"><\/p>\n<p> Dan <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d8af8ced46d93e73dc5290e0cca4dc6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{w}}.\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"17\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0c010556cb8d46303e7253102ef28e47_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{x}} = (4,2,5)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"88\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-88611544e069c7a373363f2f708dcd8c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} = (1,-1,0) \\qquad \\vv{\\text{v}} = (1,2,2) \\qquad \\vv{\\text{w}} = (-1,1,-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"383\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E4F0FE\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E4F0FE\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Lihat solusinya<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Sehingga vektornya<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-88e41d561c3898029b7b94d7014c1e79_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{x}}\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> merupakan kombinasi linier dari vektor-vektor lainnya, maka persamaan berikut harus dipenuhi:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-06d3d6ec5ca4921b109f8f974e73cbbd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a_1\\vv{\\text{u}}+a_2\\vv{\\text{v}}+ a_3\\vv{\\text{w}}=\\vv{\\text{x}}\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"160\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Oleh karena itu kami mengganti setiap vektor dengan koordinatnya:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1b5da9716a3ae4f55bf8997927615f71_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a_1\\begin{pmatrix} 1 \\\\-1\\\\0 \\end{pmatrix}+a_2\\begin{pmatrix} 1 \\\\2\\\\2 \\end{pmatrix}+ a_3\\begin{pmatrix} -1 \\\\1\\\\-1 \\end{pmatrix}=\\begin{pmatrix} 4 \\\\2\\\\5 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"310\" style=\"vertical-align: -27px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kami mengalikan setiap vektor dengan konstanta:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ea9db980d051c022dc56036cd96b054f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix} a_1 \\\\-a_1\\\\0 \\end{pmatrix}+\\begin{pmatrix} a_2 \\\\2a_2\\\\2a_2 \\end{pmatrix}+ \\begin{pmatrix} -a_3 \\\\a_3\\\\-a_3 \\end{pmatrix}=\\begin{pmatrix} 4 \\\\2\\\\5 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"278\" style=\"vertical-align: -27px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kami menambahkan vektor:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8e0fc02c135530814884b62685cc22b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix} a_1 +a_2-a_3\\\\-a_1+2a_2+a_3\\\\ 2a_2-a_3 \\end{pmatrix}=\\begin{pmatrix} 4 \\\\2\\\\5 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"202\" style=\"vertical-align: -27px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Oleh karena itu kami memperoleh sistem persamaan berikut:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1ea3ca998fc7d9d9b2cf42d43a5bf0a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left. \\begin{array}{l} a_1 +a_2-a_3 = 4 \\\\[2ex] -a_1+2a_2+a_3 =2\\\\[2ex] 2a_2-a_3 = 5 \\end{array} \\right\\}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"171\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kami memecahkan sistem yang diperoleh dengan metode Gauss: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c808441bc71bd26e333ebe2169b738ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left( \\begin{array}{ccc|c} 1&amp;1&amp;-1&amp; 4 \\\\[2ex] -1&amp;2&amp;1&amp;2\\\\[2ex] 0&amp;2&amp;-1&amp;5 \\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"149\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-941792a2de155bc284b14e34dc561418_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left(\\begin{array}{ccc|c} 1&amp;1&amp;-1&amp; 4 \\\\[2ex] -1&amp;2&amp;1&amp;2\\\\[2ex] 0&amp;2&amp;-1&amp;5 \\end{array} \\right) \\begin{array}{c} \\\\[2ex] \\xrightarrow{F_2+F_1}\\\\[2ex] &amp; \\end{array} \\left( \\begin{array}{ccc|c} 1&amp;1&amp;-1&amp; 4 \\\\[2ex] 0&amp;3&amp;0&amp;6\\\\[2ex] 0&amp;2&amp;-1&amp;5\\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"369\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7105de2fa579f40818bccc2df48961ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left(\\begin{array}{ccc|c} 1&amp;1&amp;-1&amp; 4 \\\\[2ex] 0&amp;3&amp;0&amp;6\\\\[2ex] 0&amp;2&amp;-1&amp;5 \\end{array} \\right) \\begin{array}{c} \\\\[2ex] \\\\[2ex] \\xrightarrow{3F_3-2F_2} \\end{array} \\left( \\begin{array}{ccc|c} 1&amp;1&amp;-1&amp; 4 \\\\[2ex] 0&amp;3&amp;0&amp;6\\\\[2ex] 0&amp;0&amp;-3&amp;3\\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"369\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Oleh karena itu, sistem langkah-langkah yang diperoleh adalah:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bfd5b2d564f66cd225c1a5987241ba14_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left. \\begin{array}{r} a_1 +a_2-a_3 = 4 \\\\[2ex] 3a_2 =6\\\\[2ex] -3a_3 = 3 \\end{array} \\right\\}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"148\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Yang harus kita lakukan sekarang adalah mengklarifikasi hal-hal yang tidak diketahui dan menemukan nilainya. Jadi dari persamaan terakhir kita temukan <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a5e5ed86162a9b0324b8f44dc16fcbce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a_3:\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"26\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-667fa5894272768e2e53f618a9752611_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-3a_3 = 3\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"71\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9b4234a97996e589d5d34b629a19bd0c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a_3 = \\cfrac{3}{-3} = -1\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"111\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dari persamaan kedua sistem, kita menghitung nilai <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2789190f1df15f5bd570b643d9bb29f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a_2:\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"26\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-45078dcd57cac62db8e98338a22dd939_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3a_2=6\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"59\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0580c5be6b3c77cbd727adef2f128343_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a_2=\\cfrac{6}{3} = 2\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"83\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan akhirnya, dari persamaan pertama sistem langkah, kita menemukan hal yang tidak diketahui <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c80696de686104689a20cb70c0033830_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a_1:\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"26\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1db29b41da87b5381698bd496ad4887e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a_1 +a_2-a_3 = 4 \\ \\xrightarrow{a_3\\ = \\ -1 \\ ; \\ a_2 \\ = \\ 2 } \\ a_1 +2-(-1) = 4\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"411\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b02c7b15b3b51ac99fe4d36f6f084283_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a_1 = 4-2-1\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"110\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d561c23489e6cc9b0680dbe0601babbf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a_1 = 1\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"49\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Oleh karena itu, penyelesaian sistem persamaan linear adalah:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a770689380f00a654857e19b755a1dd6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a_1=1 \\qquad a_2=2 \\qquad a_3 = -1\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"233\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Jadi vektornya<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-88e41d561c3898029b7b94d7014c1e79_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{x}}\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> Hal ini dapat dinyatakan dengan kombinasi linier berikut: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-80aba06b670bf9eedd4074be0750c3d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{x}}= a_1\\vv{\\text{u}}+a_2\\vv{\\text{v}}+ a_3\\vv{\\text{w}}\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"160\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7115a844fd089e1dd6d17e0148dfe115_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{x}}= 1\\vv{\\text{u}}+2\\vv{\\text{v}}-1\\vv{\\text{w}}\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"135\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5042840d8d9f0844c2f122aa96f850a6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\mathbf{x}}\\bm{= }\\vv{\\mathbf{u}}\\bm{+} \\bm{2} \\vv{\\mathbf{v}} \\bm{-} \\vv{\\mathbf{w}}\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"98\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Latihan 3<\/h3>\n<p> Ekspresikan vektornya<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-88e41d561c3898029b7b94d7014c1e79_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{x}}\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> sebagai kombinasi linier vektor<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4b6be5a59bbf478047e4f3ace338ee48_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}}, \\vv{\\text{v}}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: -4px;\"><\/p>\n<p> Dan <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d8af8ced46d93e73dc5290e0cca4dc6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{w}}.\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"17\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e87fcd25b965f26fff25c11b2c341f5e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{x}} = (-1,5,-3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"116\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f4d916d955d40ff456668de002eebc5e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} = (1,3,-1) \\qquad \\vv{\\text{v}} = (2,-3,-2) \\qquad \\vv{\\text{w}} = (0,-2,1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"397\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E4F0FE\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E4F0FE\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Lihat solusinya<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Kami mengusulkan persamaan kombinasi linier terhadap vektor <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-910bbc90f3e6b9fb743fe6e64dbb83d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{x}} :\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"18\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-06d3d6ec5ca4921b109f8f974e73cbbd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a_1\\vv{\\text{u}}+a_2\\vv{\\text{v}}+ a_3\\vv{\\text{w}}=\\vv{\\text{x}}\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"160\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Oleh karena itu kami mengganti setiap vektor dengan komponen-komponennya:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6c8f5b0f83b3724f96bea45f4f8c6770_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a_1\\begin{pmatrix} 1 \\\\3\\\\-1 \\end{pmatrix}+a_2\\begin{pmatrix} 2 \\\\-3\\\\-2 \\end{pmatrix}+ a_3\\begin{pmatrix} 0 \\\\-2\\\\1 \\end{pmatrix}=\\begin{pmatrix} -1 \\\\5\\\\-3 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"337\" style=\"vertical-align: -27px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kami mengalikan setiap vektor dengan masing-masing vektor yang tidak diketahui:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-66170c955f7d70bd675d864ad5f346a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix} a_1 \\\\3a_1\\\\-a_1 \\end{pmatrix}+\\begin{pmatrix} 2a_2 \\\\ -3a_2\\\\ -2a_2 \\end{pmatrix}+ \\begin{pmatrix} 0 \\\\-2a_3\\\\a_3 \\end{pmatrix}=\\begin{pmatrix} -1 \\\\5\\\\-3 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"314\" style=\"vertical-align: -27px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kami melakukan penambahan vektor:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e2a60cf7c088c8640c23e6c86ed1c00d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix} a_1 +2a_2\\\\3a_1-3a_2-2a_3\\\\ -a_1-2a_2+a_3 \\end{pmatrix}=\\begin{pmatrix} -1 \\\\5\\\\-3 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"220\" style=\"vertical-align: -27px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Oleh karena itu kami memperoleh sistem persamaan berikut:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-acdcf13a945bca16684be340d27e3523_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left. \\begin{array}{l} a_1 +2a_2 = -1 \\\\[2ex] 3a_1-3a_2-2a_3 =5\\\\[2ex] -a_1-2a_2+a_3 = -3 \\end{array} \\right\\}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"185\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kami memecahkan sistem yang diperoleh dengan metode Gauss: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e49ae26fc68a865214bd9b6146b7aa99_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left( \\begin{array}{ccc|c} 1&amp;2&amp;0&amp; -1 \\\\[2ex] 3&amp;-3&amp;-2&amp;5\\\\[2ex] -1&amp;-2&amp;1&amp;-3 \\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"177\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e4c56b420242d0abe6f77b3ed1a60e54_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left(\\begin{array}{ccc|c} 1&amp;2&amp;0&amp; -1 \\\\[2ex] 3&amp;-3&amp;-2&amp;5\\\\[2ex] -1&amp;-2&amp;1&amp;-3 \\end{array} \\right) \\begin{array}{c} \\\\[2ex] \\xrightarrow{F_2-3F_1}\\\\[2ex] \\xrightarrow{F_3+F_1} \\end{array} \\left( \\begin{array}{ccc|c} 1&amp;2&amp;0&amp; -1 \\\\[2ex] 0&amp;-9&amp;-2&amp;8\\\\[2ex] 0&amp;0&amp;1&amp;-4\\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"431\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">Oleh karena itu, sistem langkah-langkah yang diperoleh adalah:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-03461ed9ebda463d2f0a1bb6894657be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left. \\begin{array}{r} a_1 +2a_2 = -1 \\\\[2ex] -9a_2-2a_3 =8\\\\[2ex] a_3 = -4 \\end{array} \\right\\}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"140\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Yang harus kita lakukan sekarang adalah mengklarifikasi hal-hal yang tidak diketahui dan menemukan nilainya. Jadi dari persamaan terakhir kita temukan <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a5e5ed86162a9b0324b8f44dc16fcbce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a_3:\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"26\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0abc9e623042fbe70cd55d4084945584_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a_3 = -4\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"64\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dari persamaan kedua sistem, kita mencari nilai <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2789190f1df15f5bd570b643d9bb29f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a_2:\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"26\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6bb0b04bcb9cce3edf56853f8b035b69_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-9a_2-2a_3 =8 \\ \\xrightarrow{a_3 \\ = \\ -4} \\ -9a_2-2\\cdot (-4) = 8\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"357\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c798313fd76263436ded44def0ac8ba5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-9a_2+8 = 8\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"102\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-dc1521b27dddd5c037002d19dbe60aa6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-9a_2 = 8-8\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"102\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e08abc2b86a9f1cc85f4da3e70f35532_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-9a_2 = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"71\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4f389c942cdaca52620cd707a732d2d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a_2=\\cfrac{0}{-9} = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"98\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan akhirnya, dari persamaan pertama sistem langkah, kita menyelesaikan hal yang tidak diketahui <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c80696de686104689a20cb70c0033830_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a_1:\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"26\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9a77d42eebe2f101d7b1e88fce265b36_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a_1 +2a_2 = -1 \\ \\xrightarrow{a_2 \\ = \\ 0 } \\ a_1=-1\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"249\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Oleh karena itu, penyelesaian sistem persamaan linear adalah:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d3d20ab34707d782258ff1df42a5a843_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a_1=-1 \\qquad a_2=0 \\qquad a_3 = -4\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"248\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Jadi vektornya<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-88e41d561c3898029b7b94d7014c1e79_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{x}}\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> dapat dinyatakan dengan menggabungkan vektor-vektor lainnya secara linear: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-80aba06b670bf9eedd4074be0750c3d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{x}}= a_1\\vv{\\text{u}}+a_2\\vv{\\text{v}}+ a_3\\vv{\\text{w}}\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"160\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c008e155198c2dd0d0e6beadda92f677_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{x}}= -1\\vv{\\text{u}}+0\\vv{\\text{v}}-4\\vv{\\text{w}}\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"149\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8327b18d65318a6d15255b12ac67aa82_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\mathbf{x}}\\bm{= -}\\vv{\\mathbf{u}}\\bm{-4} \\vv{\\mathbf{w}}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"92\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Latihan 4<\/h3>\n<p> Tentukan apakah vektornya<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-88e41d561c3898029b7b94d7014c1e79_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{x}}\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> dapat dinyatakan sebagai kombinasi linier dari vektor-vektor<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4b6be5a59bbf478047e4f3ace338ee48_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}}, \\vv{\\text{v}}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: -4px;\"><\/p>\n<p> Dan<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d8af8ced46d93e73dc5290e0cca4dc6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{w}}.\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"17\" style=\"vertical-align: 0px;\"><\/p>\n<p> Dalam hal ini, temukan ekspresi yang menghubungkannya. <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-39c6f0a533d9bb15483b3ee9bbd2b1cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{x}} = (2,1,-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"102\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e03e61028e9e49d640d0702e0ee056e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} = (3,-1,1) \\qquad \\vv{\\text{v}} = (-1,2,0) \\qquad \\vv{\\text{w}} = (1,3,1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"369\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E4F0FE\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E4F0FE\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Lihat solusinya<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Sehingga vektornya<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-88e41d561c3898029b7b94d7014c1e79_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{x}}\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> merupakan kombinasi linier dari vektor-vektor lainnya, maka persamaan berikut harus dipenuhi:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-06d3d6ec5ca4921b109f8f974e73cbbd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a_1\\vv{\\text{u}}+a_2\\vv{\\text{v}}+ a_3\\vv{\\text{w}}=\\vv{\\text{x}}\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"160\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Oleh karena itu kami mengganti setiap vektor dengan koordinatnya:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-649abb0a558488a33e4f1e89d952dbf4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a_1\\begin{pmatrix} 3 \\\\-1\\\\1 \\end{pmatrix}+a_2\\begin{pmatrix} -1 \\\\2\\\\0 \\end{pmatrix}+ a_3\\begin{pmatrix} 1 \\\\3\\\\1 \\end{pmatrix}=\\begin{pmatrix} 2 \\\\1\\\\-1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"323\" style=\"vertical-align: -27px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kami mengalikan setiap vektor dengan koefisiennya:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e555b6f0b4b201e2678bd843d6924f0b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix} 3a_1 \\\\-a_1\\\\a_1 \\end{pmatrix}+\\begin{pmatrix} -a_2 \\\\2a_2\\\\0 \\end{pmatrix}+ \\begin{pmatrix} a_3 \\\\3a_3\\\\a_3 \\end{pmatrix}=\\begin{pmatrix} 2 \\\\1\\\\-1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"292\" style=\"vertical-align: -27px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kami menambahkan vektor:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c3437e5ddbc157f4471e2a6524f0f5ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix} 3a_1 -a_2+a_3\\\\-a_1+2a_2+3a_3\\\\ a_1+a_3 \\end{pmatrix}=\\begin{pmatrix} 2 \\\\1\\\\-1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"225\" style=\"vertical-align: -27px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Oleh karena itu, ekspresi sebelumnya setara dengan sistem persamaan berikut:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f51b7e801b8314c51b983f1f24be15e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left. \\begin{array}{l} 3a_1 -a_2+a_3 = 2 \\\\[2ex] -a_1+2a_2+3a_3 =1\\\\[2ex] a_1+a_3 = -1 \\end{array} \\right\\}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"180\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kami sekarang menyelesaikan sistem yang diperoleh dengan metode Gauss: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-031b14d5aca6a41d897ca575440b1197_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left( \\begin{array}{ccc|c} 3&amp;-1&amp;1&amp; 2 \\\\[2ex] -1&amp;2&amp;3&amp;1\\\\[2ex] 1&amp;0&amp;1&amp;-1 \\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"163\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2caf1e1104b8b67e13d452bbd20d13b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left(\\begin{array}{ccc|c} 3&amp;-1&amp;1&amp; 2 \\\\[2ex] -1&amp;2&amp;3&amp;1\\\\[2ex] 1&amp;0&amp;1&amp;-1 \\end{array} \\right) \\begin{array}{c} \\\\[2ex] \\xrightarrow{3F_2+F_1}\\\\[2ex] \\xrightarrow{3F_3-F_1} \\end{array} \\left( \\begin{array}{ccc|c} 3&amp;-1&amp;1&amp; 2 \\\\[2ex] 0&amp;5&amp;10&amp;5\\\\[2ex] 0&amp;1&amp;2&amp;-5\\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"412\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d4deec2426c0b9bb0b8e8a3d95155fd9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left(\\begin{array}{ccc|c} 3&amp;-1&amp;1&amp; 2 \\\\[2ex] 0&amp;5&amp;10&amp;5\\\\[2ex] 0&amp;1&amp;2&amp;-5 \\end{array} \\right) \\begin{array}{c} \\\\[2ex] \\\\[2ex] \\xrightarrow{5F_3-F_2} \\end{array} \\left( \\begin{array}{ccc|c}3&amp;-1&amp;1&amp; 2 \\\\[2ex] 0&amp;5&amp;10&amp;5\\\\[2ex] 0&amp;0&amp;0&amp;-30\\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"416\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Oleh karena itu kami memperoleh sistem persamaan berikut:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e537d5c481ceedeaebf95334d72199ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left. \\begin{array}{r} 3a_1 -a_2+a_3 = 2 \\\\[2ex] 5a_2 +10a_3=5\\\\[2ex] 0 = -30 \\end{array} \\right\\}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"157\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Namun, persamaan terakhir tidak akan pernah bisa dipenuhi, karena 0 tidak akan pernah sama dengan -30 berapa pun nilai yang diambil dari bilangan yang tidak diketahui. Oleh karena itu, sistem tidak memiliki solusi dan ini berarti <strong>tidak ada kombinasi linier<\/strong> untuk menghitung vektor<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9f9ba5824d0d2c7ebfa020ea72dc6a11_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{x}}.\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"13\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Pada halaman ini Anda akan menemukan penjelasan tentang apa yang dimaksud dengan kombinasi linier antar vektor. Selain itu, Anda akan dapat melihat contoh bagaimana vektor dinyatakan sebagai kombinasi linier dan, sebagai tambahan, Anda akan dapat berlatih dengan latihan dan soal yang diselesaikan langkah demi langkah. Apa yang dimaksud dengan kombinasi linier vektor? Pengertian kombinasi linier &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/id\/kombinasi-linier-vektor-contoh-latihan-yang-diselesaikan\/\"> <span class=\"screen-reader-text\">Kombinasi vektor linier<\/span> Selengkapnya &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"","footnotes":""},"categories":[54],"tags":[],"class_list":["post-71","post","type-post","status-publish","format-standard","hentry","category-vektor"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Kombinasi linear vektor -<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathority.org\/id\/kombinasi-linier-vektor-contoh-latihan-yang-diselesaikan\/\" \/>\n<meta property=\"og:locale\" content=\"id_ID\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Kombinasi linear vektor -\" \/>\n<meta property=\"og:description\" content=\"Pada halaman ini Anda akan menemukan penjelasan tentang apa yang dimaksud dengan kombinasi linier antar vektor. Selain itu, Anda akan dapat melihat contoh bagaimana vektor dinyatakan sebagai kombinasi linier dan, sebagai tambahan, Anda akan dapat berlatih dengan latihan dan soal yang diselesaikan langkah demi langkah. Apa yang dimaksud dengan kombinasi linier vektor? 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