Resultado de API de MediaWiki
Esta é a representación HTML do formato JSON. O HTML é bo para depurar, pero non é axeitado para usar nunha aplicación.
Especifica o parámetro format para cambiar o formato de saída. Para ver a representación non HTML do formato JSON, define format=json.
Consulta a documentación completa ou a axuda da API para obter máis información.
{
"compare": {
"fromid": 1,
"fromrevid": 1,
"fromns": 4,
"fromtitle": "Testwiki:Zona de probas",
"toid": 2,
"torevid": 2,
"tons": 0,
"totitle": "\u00c1lxebra Lineal: Produto interno",
"*": "<tr><td colspan=\"2\" class=\"diff-lineno\" id=\"mw-diff-left-l1\">Li\u00f1a 1:</td>\n<td colspan=\"2\" class=\"diff-lineno\">Li\u00f1a 1:</td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">A </del>'''<del class=\"diffchange diffchange-inline\">zona de probas</del>''' \u00e9 <del class=\"diffchange diffchange-inline\">para realizar experimentos </del>de <del class=\"diffchange diffchange-inline\">edici\u00f3n e visualizar os resultados no '''Wikibooks en galego'''. Pode facer libremente cantas probas estime oportuno. Ningu\u00e9n lle vai chamar a atenci\u00f3n polos cambios que faga nesta p\u00e1xina</del>.</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{Template:\u00c1lxebra}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\"><br clear=\"both\"></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">En [[\u00c1lxebra Lineal]], chamamos </ins>'''<ins class=\"diffchange diffchange-inline\">produto interno</ins>''' <ins class=\"diffchange diffchange-inline\">a unha funci\u00f3n de dous vectores que satisfai determinados axiomas. O [[produto escalar]], comumente usado na [[xeometr\u00eda euclidiana]], </ins>\u00e9 <ins class=\"diffchange diffchange-inline\">un caso especial </ins>de <ins class=\"diffchange diffchange-inline\">produto interno</ins>.</div></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">As edici\u00f3ns ou comentarios que se po\u00f1an nesta p\u00e1xina non son permanentes ou \u00fanico que permanece \u00e9 este texto introductorio. Cada certo tempo borrase o contido da '''zona de probas'''.</del></div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">==Definici\u00f3n==</ins></div></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">Premendo en '''editar''' podese cambiar o contido</del>. <del class=\"diffchange diffchange-inline\">Na opci\u00f3n '''Amosa o previo''' obtemos unha previsualizaci\u00f3n da edici\u00f3n. Con '''Garda </del>a <del class=\"diffchange diffchange-inline\">p\u00e1xina'''</del>, <del class=\"diffchange diffchange-inline\">queda gardado o resultado e rematou a edici\u00f3n. E unha boa pr\u00e1ctica escribir en '''Resumo''' un pequeno comentario das probas realizadas.</del></div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Sexa \u2018\u2018\u2018V\u2019\u2018\u2018 un [[espazo vectorial]] sobre un [[corpo_(matem\u00e1tica)|corpo]] \u2018\u2018\u2018K\u2019\u2018\u2018</ins>. <ins class=\"diffchange diffchange-inline\">En \u2018\u2018\u2018V\u2019\u2018\u2018, p\u00f3dese definir </ins>a <ins class=\"diffchange diffchange-inline\">[[funci\u00f3n]] binaria <math>\\langle \\cdot</ins>,<ins class=\"diffchange diffchange-inline\">\\cdot\\rangle: V \\times V \\rightarrow K</math> (denominada \u2018\u2018\u2018produto interno\u2019\u2018\u2018), que satisfai os seguintes axiomas:</ins></div></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange\">: <math>\\langle u,v\\rangle\u00a0 = \\overline{\\langle v,u\\rangle }</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange\">: <math>\\langle u+v, w\\rangle\u00a0 = \\langle u,w\\rangle\u00a0 + \\langle v,w\\rangle</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange\">: <math>\\langle \\lambda u, v\\rangle\u00a0 = \\lambda \\langle u, v\\rangle</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange\">: Se <math>v \\ne 0</math>, ent\u00f3n <math>\\langle v, v\\rangle </math>&gt; <math>0</math></ins></div></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">----</del></div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">en que \u2018\u2018u\u2019\u2018, \u2018\u2018v\u2019\u2018 e \u2018\u2018w\u2019\u2018 son vectores </ins>de <ins class=\"diffchange diffchange-inline\">\u2018\u2018\u2018V\u2019\u2018\u2018, e \u2018\u2018&lambda;\u2019\u2018 \u00e9 un elemento de \u2018\u2018\u2018K\u2019\u2018\u2018.</ins></div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">'''A partir </del>de <del class=\"diffchange diffchange-inline\">aqu\u00ed pode facer as s\u00faas probas:'''</del></div></td><td colspan=\"2\" class=\"diff-side-added\"></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">----</del></div></td><td colspan=\"2\" class=\"diff-side-added\"></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">== Unha proba feita por min ==</del></div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">A partir deses axiomas, \u00e9 pos\u00edbel probar as seguintes consecuencias:</ins></div></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">Esta \u00e9 unha proba para ver como vai isto. Quero ver unha \u00f1 escrita</del>, <del class=\"diffchange diffchange-inline\">unha acentuaci\u00f3n e m\u00e1is un ping\u00fcino.</del></div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">: <math>\\langle u</ins>, <ins class=\"diffchange diffchange-inline\">v+w\\rangle\u00a0 = \\langle u, v\\rangle\u00a0 + \\langle u, w\\rangle</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">: <math>\\langle u, \\lambda v\\rangle\u00a0 = \\overline{\\lambda}\\langle u,v\\rangle</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">: Se <math>v = 0</math>, ent\u00f3n <math>\\langle v, v\\rangle\u00a0 = 0</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">: Se <math>\\langle v, v\\rangle\u00a0 = 0</math>, ent\u00f3n <math>v = 0</math></ins></div></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">Vexamos como vai o tema das ecuaci\u00f3ns:</del></div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">===Exemplos===</ins></div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\"><math>\\int_{-\\infty}^{\\infty} \\frac{1}{\\sqrt{2 \\pi}} \\exp(-x^2/2) dx</math></del></div></td><td colspan=\"2\" class=\"diff-side-added\"></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">Vou ver como queda ....</del></div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">O [[produto escalar]] sobre o espazo vectorial <math>\\mathbb{R}^3</math> satisfai os axiomas do produto interno e def\u00ednese por:</ins></div></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">Agora vou probar coas \u00a1admiraci\u00f3ns!</del></div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:<math>\\langle (x_1, y_1, z_1), (x_2, y_2, z_2)\\rangle = x_1x_2 + y_1y_2 + z_1z_2</math></ins></div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">\u00bfQuedar\u00e1 ben?</del></div></td><td colspan=\"2\" class=\"diff-side-added\"></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange\">Se \u2018\u2018f\u2019\u2018 e \u2018\u2018g\u2019\u2018 son d\u00faas funci\u00f3ns, \u00e9 pos\u00edbel definir o produto interno:</ins></div></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange\">:<math> \\langle f, g \\rangle = \\int f(x)\\overline{g(x)}\\,dx</math></ins></div></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">Novas probas</del></div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">==Vetores ortogonais==</ins></div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">hijo</del></div></td><td colspan=\"2\" class=\"diff-side-added\"></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">jbkhvj</del></div></td><td colspan=\"2\" class=\"diff-side-added\"></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">[[Category:Ecuaci\u00f3n|Zona de probas</del>, <del class=\"diffchange diffchange-inline\">Wikibooks:]]</del></div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Dise que dous vectores <math>u</ins>, <ins class=\"diffchange diffchange-inline\">v \\in V</math> son \u2018\u2018\u2018ortogonais\u2019\u2018\u2018 se <math>\\langle u, v\\rangle\u00a0 = 0</math>.</ins></div></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><!--<del class=\"diffchange diffchange-inline\">Por favor</del>, non <del class=\"diffchange diffchange-inline\">borrar </del>a <del class=\"diffchange diffchange-inline\">categor\u00eda</del>--></div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Consecuencias:</ins></div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">[[Category</del>:<del class=\"diffchange diffchange-inline\">Axuda|Zona </del>de <del class=\"diffchange diffchange-inline\">probas</del>, <del class=\"diffchange diffchange-inline\">Wikibooks</del>:]]</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:Se <math>\\langle u, v\\rangle = 0, \\forall v \\in V</math>, ent\u00f3n <math>u = 0</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:Se <math>\\langle T(u), v\\rangle = 0, \\forall u,v \\in V</math>, ent\u00f3n <math>T = 0</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">==Complemento ortogonal==</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Sexa <math>v \\in V, v \\ne 0</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Def\u00ednese o complemento ortogonal de \u2018\u2018v\u2019\u2018, <math>v^\\perp</math>, como:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:<math>v^\\perp = \\{ v \\}^\\perp = \\{ u \\in V\u00a0 | \\langle u, v \\rangle = 0 \\}. </math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Consecuencias:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:<math>v^\\perp</math> \u00e9 un subespazo vectorial de V</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:Sexa <math>W</math> un subespazo vectorial de V, e <math>\\alpha = \\{v_1, v_2, \\ldots, v_n\\}</math> unha base de <math>W</math>. <math>v \\in W^\\perp \\iff v \\in v_i^\\perp, i = 1, \\ldots, n</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:<math>(W^\\perp)^\\perp = W</math>, W \u00e9 subespazo de V.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">==Norma==</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Sexa \u2018\u2018V\u2019\u2018 un espazo vectorial sobre o corpo \u2018\u2018K\u2019\u2018, con produto interno.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Def\u00ednese a \u2018\u2018\u2018norma\u2019\u2018\u2018 ou \u2018\u2018\u2018lonxitude\u2019\u2018\u2018 dun vector <math>v \\in V</math> como sendo o n\u00famero <math>\\sqrt{\\langle v, v \\rangle}</math>, que indicamos por <math>|v|</math>.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Consecuencias:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:<math>|v| = 0 \\Longleftrightarrow v = 0</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:Se <math>v \\ne 0</math>, ent\u00f3n <math>|v| > 0</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:<math>|\\lambda v| = |\\lambda| |v|, \\forall \\lambda \\in K, v \\in V</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:Se <math>\\langle u, v \\rangle = 0</math>, ent\u00f3n <math>|u + v|^2 = |u|^2 + |v|^2</math> (Teorema de Pit\u00e1goras)</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">==Proxeci\u00f3n ortogonal==</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">===Proxeci\u00f3n dun vector v na direcci\u00f3n dun vector u, en que u &ne; 0===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Def\u00ednese \u2018\u2018\u2018esa\u2019\u2018\u2018 proxeci\u00f3n como sendo o vector</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\"><math>\\mbox{prox}_uv = \\frac{\\langle v, u \\rangle}{\\langle u, u \\rangle} \\cdot u</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">===Proxeci\u00f3n dun vector \u2018\u2018v\u2019\u2018 sobre un subespazo vectorial \u2018\u2018W\u2019\u2018 de \u2018\u2018V\u2019\u2018===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Sexa <math>W = [u_1, u_2]</math>, en que <math> \\{ u_1, u_2 \\}</math> \u00e9 unha base ortogonal de \u2018\u2018W\u2019\u2018.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\"><math>\\mbox{prox}_Wv = \\mbox{prox}_{u_1}v + \\mbox{prox}_{u_2}v</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">==Desigualdade de Cauchy-Schwarz==</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Dados <math>u,v \\in V</math>, ent\u00f3n <math>| \\langle u, v \\rangle | \\le |u| \\cdot |v|</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">==Desigualdade triangular==</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\"><math> |u + v| \\le |u| + |v|, \\forall u, v \\in V</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">==Base ortogonal e ortonormal==</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Unha base <math> \\{ v_1, v_2, \\ldots, v_n \\} </math> de V \u00e9 dita ortonormal se <math> \\langle v_i, v_x \\rangle = \\delta ix</math>, en que </ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:<math>\\delta ix = 1</math>, se i = x</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:<math>\\delta ix = 0</math>, se i &ne; x</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><!--</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\"><math> \\delta ix = \\left\\{ \\begin{array}{ll} 1, & \\mbox{se } i=x \\\\ 0</ins>, <ins class=\"diffchange diffchange-inline\">& \\mbox{se } i \\ne x\\end{array}\\right.</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">--></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">A base \u00e9 ortogonal se os vectores son ortogonais dous a dous.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Propiedade: \u2018\u2018n\u2019\u2018 vectores </ins>non<ins class=\"diffchange diffchange-inline\">-nulos e ortogonais dous a dous, nun espazo de dimensi\u00f3n \u2018\u2018n\u2019\u2018, son linearmente independentes.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">==Proceso de ortogonalizaci\u00f3n de Gram-Schmidt==</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Dada unha base <math> \\{ v_1, v_2, \\ldots, v_n \\} </math> de V, podemos atopar, </ins>a <ins class=\"diffchange diffchange-inline\">partir desta base,</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">unha base ortogonal <math> \\{ u_1, u_2, \\ldots, u_n \\} </math> de V.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\"><math> u_i = v_i </ins>- <ins class=\"diffchange diffchange-inline\">\\sum_{k=1}^{i</ins>-<ins class=\"diffchange diffchange-inline\">1}\u00a0 \\frac{ \\langle v_i, u_k \\rangle }{ \\langle u_k, u_k \\rangle } u_k </math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">==Distancia entre dous vectores==</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Def\u00ednese a distancia entre dous vectores calquera, \u2018\u2018u\u2019\u2018 e \u2018\u2018v\u2019\u2018, como sendo</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\"><math>d(u,v) = |u - v| </math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Unha funci\u00f3n distancia ten as seguintes propiedades:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:<math>d(u, v) \\xe 0</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:<math>\\quad d(u, v) = 0 \\Leftrightarrow u = v</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:<math>d(u, v) = d(v, u)</math</ins>></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>:<ins class=\"diffchange diffchange-inline\"><math>d(u,v) \\le d(u, w) + d(w, v)</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Tales propiedades poden ser facilmente verificadas pola definici\u00f3n de norma.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">==Mellor aproximaci\u00f3n dun vector v de V por un vector de W, subespazo vectorial </ins>de <ins class=\"diffchange diffchange-inline\">V==</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Se <math>d(v, u) \\le d(v, u'), \\forall u' \\in W</math></ins>, <ins class=\"diffchange diffchange-inline\">ent\u00f3n u \u00e9 o vector de W que d\u00e1 a aproximaci\u00f3n m\u00e1is adecuada de v por un vector de W.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Demostrase que <math>u = prox_W v</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{Template:\u00c1lxebra}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[Categor\u00eda</ins>:<ins class=\"diffchange diffchange-inline\">\u00c1lxebra lineal</ins>]]</div></td></tr>\n"
}
}