{"id":74,"date":"2023-09-16T13:04:23","date_gmt":"2023-09-16T13:04:23","guid":{"rendered":"https:\/\/mathority.org\/it\/esempi-di-combinazioni-lineari-di-vettori-esercizi-risolti\/"},"modified":"2023-09-16T13:04:23","modified_gmt":"2023-09-16T13:04:23","slug":"esempi-di-combinazioni-lineari-di-vettori-esercizi-risolti","status":"publish","type":"post","link":"https:\/\/mathority.org\/it\/esempi-di-combinazioni-lineari-di-vettori-esercizi-risolti\/","title":{"rendered":"Combinazione lineare di vettori"},"content":{"rendered":"<p>In questa pagina troverai la spiegazione di cosa significa una combinazione lineare tra vettori. Inoltre, potrai vedere un esempio di come si esprime un vettore come combinazione lineare e, inoltre, potrai esercitarti con esercizi e problemi risolti passo dopo passo. <\/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> Cos&#8217;\u00e8 la combinazione lineare di vettori?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> La definizione di combinazione lineare \u00e8 la seguente: <\/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\"> Una <strong>combinazione lineare<\/strong> di un insieme di vettori \u00e8 il vettore ottenuto sommando tutti i vettori dell&#8217;insieme moltiplicati per scalari (numeri reali).<\/p>\n<p style=\"text-align:left\"> In altre parole, dato un insieme di vettori<\/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> una loro combinazione lineare sarebbe:<\/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\"> Dove i coefficienti<\/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> Questi sono numeri reali.<\/p>\n<\/div>\n<p> Pertanto, un vettore che \u00e8 una combinazione lineare di altri vettori significa che il primo pu\u00f2 essere espresso in termini del secondo.<\/p>\n<p> Questo concetto pu\u00f2 essere meglio compreso rappresentando graficamente un vettore nel piano che \u00e8 una combinazione lineare di due vettori: <\/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=\"combinazione lineare di vettori in r3\" class=\"wp-image-781\" width=\"405\" height=\"408\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Come puoi vedere nella rappresentazione grafica sopra, il vettore<\/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> possono essere ottenuti da vettori<\/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> E<\/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> eseguire operazioni sui vettori. Pertanto, il vettore<\/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> \u00e8 una combinazione lineare degli altri due vettori.<\/p>\n<p> \u00c8 bene sottolineare che questa combinazione lineare \u00e8 <strong>unica<\/strong> , ovvero esiste una sola combinazione lineare ammissibile per ogni vettore. Poich\u00e9, seguendo l&#8217;esempio precedente, se moltiplicassimo<\/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> per 6 invece di 4 otterremmo un altro vettore diverso. <\/p>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-106\"><\/div>\n<\/div>\n<p> Inoltre, una delle propriet\u00e0 della combinazione lineare nel piano (in R2) \u00e8 che qualsiasi vettore pu\u00f2 essere posto come combinazione lineare di altri due vettori se hanno direzioni diverse, cio\u00e8 se non sono paralleli.<\/p>\n<p> Inoltre, a volte possiamo identificare a occhio che due vettori sono una combinazione lineare. Per fare ci\u00f2 \u00e8 sufficiente che le sue componenti siano <strong>proporzionali<\/strong> . Ad esempio, le coordinate dei seguenti due vettori sono proporzionali e, quindi, i vettori sono una combinazione lineare:<\/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> Infine, sia in uno spazio vettoriale bidimensionale (in R2) che tridimensionale (in R3), se esiste una combinazione lineare all&#8217;interno di un insieme di vettori, ci\u00f2 implica che essi sono <strong>linearmente dipendenti<\/strong> l&#8217;uno dall&#8217;altro. D&#8217;altra parte, se non \u00e8 possibile alcuna combinazione lineare tra i vettori, ci\u00f2 significa che sono <strong>linearmente indipendenti<\/strong> .<\/p>\n<p> Se quest&#8217;ultimo concetto non ti \u00e8 del tutto chiaro, ti consigliamo di consultare la nostra spiegazione dei <a href=\"https:\/\/mathority.org\/it\/vettori-indipendenti-e-linearmente-dipendenti-indipendenza-dipendenza-lineare\/\">vettori linearmente dipendenti e indipendenti<\/a> , qui troverai cosa significa che i vettori sono linearmente dipendenti o indipendenti, esempi di ogni tipo e le differenze tra loro. . Questo concetto \u00e8 molto usato e, infatti, \u00e8 chiesto molto agli esami, quindi \u00e8 importante che tu lo capisca bene. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"como-expresar-un-vector-como-combinacion-lineal-de-otros-vectores\"><\/span> Come esprimere un vettore come combinazione lineare di altri vettori <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> Vedremo poi come risolvere un tipico problema in cui ci viene chiesto di trovare la combinazione lineare di un vettore.<\/p>\n<ul>\n<li> Esprimere il vettore\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> come combinazione lineare di<\/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> E <\/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> In modo che il vettore<\/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> essere una combinazione lineare degli altri vettori, deve essere soddisfatta la seguente equazione:<\/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> Dove i coefficienti<\/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> E<\/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> Queste sono le incognite che dobbiamo trovare.<\/p>\n<p> Sostituiamo quindi ogni vettore con le sue coordinate:<\/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> Moltiplichiamo ciascun vettore per il suo coefficiente:<\/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> Aggiungiamo i vettori:<\/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> Ciascuna coordinata sinistra deve essere uguale a ciascuna coordinata destra. Abbiamo quindi 3 equazioni:<\/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> Non resta che risolvere il sistema di equazioni ottenuto. Per fare questo utilizza il metodo che preferisci (metodo di sostituzione, regola di Cramer, metodo di Gauss-Jordan, ecc.), in questo caso utilizzeremo il metodo di 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> Il sistema di passi ottenuto \u00e8 quindi:<\/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> Tutto quello che dobbiamo fare ora \u00e8 chiarire le incognite e trovarne il valore. Quindi dall&#8217;ultima equazione troviamo<\/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> Dalla seconda equazione del sistema, calcoliamo il valore di <\/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> E infine, dalla prima equazione del sistema di gradini, troviamo l&#8217;incognita<\/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> La soluzione del sistema di equazioni lineari \u00e8 quindi:<\/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> Quindi il vettore<\/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> Pu\u00f2 essere espresso dalla seguente combinazione lineare: <\/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> Esiste quindi effettivamente una dipendenza lineare tra i vettori. D&#8217;altra parte, se non fosse stata ottenuta alcuna soluzione del sistema di equazioni, ci\u00f2 significherebbe che il vettore<\/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> Esso \u00e8 linearmente indipendente rispetto agli altri vettori e, pertanto, non vi sarebbe alcuna combinazione lineare possibile per ottenere detto vettore dagli altri vettori. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-combinacion-lineal-de-vectores\"><\/span> Esercizi risolti sulla combinazione lineare di vettori<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> Esercizio 1<\/h3>\n<p> Tra i seguenti tre vettori, indicare quali coppie sono combinazioni lineari l&#8217;una dell&#8217;altra. Inoltre, trova la relazione di combinazione lineare di dette coppie di vettori. <\/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>Vedi la soluzione<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Per sapere se una coppia di vettori \u00e8 una combinazione lineare, dobbiamo vedere se le loro coordinate sono proporzionali.<\/p>\n<p class=\"has-text-align-left\"> Per prima cosa controlliamo il vettore<\/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> con il vettore<\/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\"> In secondo luogo, controlliamo il vettore<\/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> con il vettore<\/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\"> Infine, testiamo il vettore<\/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> con il vettore<\/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\"> Quindi l&#8217;unica coppia di vettori che sono combinazioni lineari lo \u00e8<\/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> E<\/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> Inoltre il loro rapporto \u00e8 il seguente:<\/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\"> O equivalente:<\/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\"> Sebbene l&#8217;affermazione non lo richieda, gli unici vettori che dipendono linearmente l&#8217;uno dall&#8217;altro lo sono<\/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> E<\/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> perch\u00e9 esiste una combinazione lineare tra loro. Le altre coppie sono linearmente indipendenti perch\u00e9 non possono essere combinate linearmente.<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Esercizio 2<\/h3>\n<p> Trova la relazione lineare tra il vettore<\/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> e l&#8217;insieme dei vettori<\/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> E <\/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>Vedi la soluzione<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> In modo che il vettore<\/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> essere una combinazione lineare degli altri vettori, deve essere soddisfatta la seguente equazione:<\/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\"> Sostituiamo quindi ogni vettore con le sue coordinate:<\/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\"> Moltiplichiamo ciascun vettore per la sua costante:<\/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\"> Aggiungiamo i vettori:<\/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\"> Otteniamo quindi il seguente sistema di equazioni:<\/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\"> Risolviamo il sistema ottenuto con il metodo di 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\"> Il sistema di passi ottenuto \u00e8 quindi:<\/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\"> Tutto quello che dobbiamo fare ora \u00e8 chiarire le incognite e trovarne il valore. Quindi dall&#8217;ultima equazione troviamo <\/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\"> Dalla seconda equazione del sistema, calcoliamo il valore di <\/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\"> E infine, dalla prima equazione del sistema di gradini, troviamo l&#8217;incognita <\/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\"> La soluzione del sistema di equazioni lineari \u00e8 quindi:<\/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\"> Quindi il vettore<\/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> Pu\u00f2 essere espresso dalla seguente combinazione lineare: <\/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\">Esercizio 3<\/h3>\n<p> Esprimere il vettore<\/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> come combinazione lineare di vettori<\/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> E <\/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>Vedi la soluzione<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Proponiamo l&#8217;equazione di combinazione lineare rispetto al vettore <\/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\"> Sostituiamo quindi ogni vettore con le sue componenti:<\/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\"> Moltiplichiamo ciascun vettore per la rispettiva incognita:<\/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\"> Eseguiamo l&#8217;addizione dei vettori:<\/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\"> Abbiamo quindi ottenuto il seguente sistema di equazioni:<\/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\"> Risolviamo il sistema ottenuto con il metodo di 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\">Il sistema di passi ottenuto \u00e8 quindi:<\/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\"> Tutto quello che dobbiamo fare ora \u00e8 chiarire le incognite e trovarne il valore. Quindi dall&#8217;ultima equazione troviamo <\/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\"> Dalla seconda equazione del sistema, troviamo il valore di <\/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\"> E infine, dalla prima equazione del sistema di passaggi, risolviamo l&#8217;incognita <\/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\"> La soluzione del sistema di equazioni lineari \u00e8 quindi:<\/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\"> Quindi il vettore<\/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> pu\u00f2 essere espresso combinando linearmente gli altri vettori: <\/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\">Esercizio 4<\/h3>\n<p> Determinare se il vettore<\/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> pu\u00f2 essere espresso come una combinazione lineare dei vettori<\/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> E<\/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> In questo caso, trova l&#8217;espressione che li collega. <\/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>Vedi la soluzione<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> In modo che il vettore<\/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> essere una combinazione lineare degli altri vettori, deve essere soddisfatta la seguente equazione:<\/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\"> Sostituiamo quindi ogni vettore con le sue coordinate:<\/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\"> Moltiplichiamo ciascun vettore per il suo coefficiente:<\/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\"> Aggiungiamo i vettori:<\/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\"> L\u2019espressione precedente equivale quindi al seguente sistema di equazioni:<\/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\"> Risolviamo ora il sistema ottenuto con il metodo di 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\"> Abbiamo quindi ottenuto il seguente sistema di equazioni:<\/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\"> Tuttavia, l\u2019ultima equazione non potr\u00e0 mai essere soddisfatta, poich\u00e9 0 non sar\u00e0 mai uguale a -30 qualunque siano i valori assunti dalle incognite. Pertanto il sistema non ha soluzione e questo implica che <strong>non esiste alcuna combinazione lineare<\/strong> per calcolare il vettore<\/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>In questa pagina troverai la spiegazione di cosa significa una combinazione lineare tra vettori. Inoltre, potrai vedere un esempio di come si esprime un vettore come combinazione lineare e, inoltre, potrai esercitarti con esercizi e problemi risolti passo dopo passo. Cos&#8217;\u00e8 la combinazione lineare di vettori? La definizione di combinazione lineare \u00e8 la seguente: Una &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/it\/esempi-di-combinazioni-lineari-di-vettori-esercizi-risolti\/\"> <span class=\"screen-reader-text\">Combinazione lineare di vettori<\/span> Leggi altro &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":[22],"tags":[],"class_list":["post-74","post","type-post","status-publish","format-standard","hentry","category-vettori"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Combinazione lineare di vettori -<\/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\/it\/esempi-di-combinazioni-lineari-di-vettori-esercizi-risolti\/\" \/>\n<meta property=\"og:locale\" content=\"it_IT\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Combinazione lineare di vettori -\" \/>\n<meta property=\"og:description\" content=\"In questa pagina troverai la spiegazione di cosa significa una combinazione lineare tra vettori. 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