- published: 26 May 2023
- views: 23955
'+pages+''); $('.stream > div:odd').addClass('bgr_color'); updateHeight('#history'); }); window.activateTabArea = ensure(function(tab, areas){ var parsed = false; var parts = (areas || '').split('/'); window.fsonload = $.inArray('fs', parts) >= 0; if(fsonload){ parts.splice(parts.indexOf('fs'), 1); } var replayMode = false; if($.inArray('replay', parts)>=0){ replayMode = 'replay'; } var noSoundMode = false; if($.inArray('nosound', parts)>=0){ noSoundMode = 'nosound'; } if($.inArray('ns', parts)>=0){ noSoundMode = 'ns'; } var previewMode = null; if($.inArray('p', parts)>=0){ previewMode = 'p'; } if($.inArray('preview', parts)>=0){ previewMode = 'preview'; } if($.inArray('repeat', parts)>=0){ replayMode = 'repeat'; } if($.inArray('r', parts)>=0 || $.inArray('ro', parts)>=0){ replayMode = 'r'; } if(replayMode){ parts.splice(parts.indexOf(replayMode), 1); } if(noSoundMode){ parts.splice(parts.indexOf(noSoundMode), 1); } if(previewMode){ parts.splice(parts.indexOf(previewMode), 1); } if(previewMode){ if(!parts.length){ parts = ['1-14', '999:59']; } } var area = parts[0]; if(tab == 'history' && false){ var page = parseInt(area || '1') || 1; $.ajax({ url: 'https://login.wn.com/recent/json/?pp='+history_pp+'&skip='+history_pp*(page-1), dataType: 'jsonp', success: function(response){ $ensure(function(){ renderHistory(response, page); }); } }); return true; } if(tab == 'global_history' && false){ var page = parseInt(area || '1') || 1; globalHistory.fetchStream(page, '', function(){ updateHeight('#global_history'); }); return true; } if(tab == 'my_playlists' && false){ var page = parseInt(area || '1') || 1; myPlaylists.fetchStream(page, '', function(){ updateHeight('#my_playlists'); }); return true; } if(tab == 'my_videos' && false){ var page = parseInt(area || '1') || 1; myVideos.fetchStream(page, '', function(){ updateHeight('#my_videos'); }); return true; } if(tab == 'related_sites' && areas && matchPosition(areas)){ var seconds = parsePosition(areas); scrollRelated(seconds); return false; } if(matchPosition(area) || matchAction(area)){ parts.unshift('1'); area = parts[0]; } if(tab == 'expand' && area && area.match(/\d+/)) { var num = parseInt(area); if(num < 100){ //FIX ME. Load news page with ajax here } else if(num > 1900){ //FIX ME. Load timeline page with ajax here } } else if(tab.match(/^playlist\d+$/)){ var playerId = parseInt(tab.substring(8)); var vp = videoplayers[playerId]; window.descriptionsholder = $('.descriptionsplace'); if(!vp) return; // why? no player? if(replayMode){ $('.replaycurrent'+playerId).attr('checked', true); vp.setReplayCurrent(true); } var playQueue = []; window.playQueue = playQueue; var playQueuePosition = 0; var playShouldStart = null; var playShouldStop = null; var parseList = function(x){ var items = x.split(/;|,/g); var results = []; for (i in items){ try{ var action = parseAction(vp, items[i]); if(!action.video){ if(window.console && console.log) console.log("Warning: No video for queued entry: " + items[i]); }else{ results.push(action); } }catch(e){ if(window.console && console.log) console.log("Warning: Can''t parse queue entry: " + items[i]); } } return results; }; var scrollToPlaylistPosition = function(vp){ var ppos = vp.getPlaylistPosition(); var el = vp.playlistContainer.find('>li').eq(ppos); var par = el.closest('.playlist_scrollarea'); par.scrollTop(el.offset().top-par.height()/2); } var updateVolumeState = function(){ if(noSoundMode){ if(noSoundMode == 'turn-on'){ clog("Sound is on, vsid="+vp.vsid); vp.setVolumeUnMute(); noSoundMode = false; }else{ clog("Sound is off, vsid="+vp.vsid); vp.setVolumeMute(); noSoundMode = 'turn-on'; } } } var playQueueUpdate = function(){ var playPosition = playQueue[playQueuePosition]; vp.playFromPlaylist(playPosition.video); scrollToPlaylistPosition(vp); playShouldStart = playPosition.start; playShouldStop = playPosition.stop; }; var playQueueAdvancePosition = function(){ clog("Advancing play position..."); playQueuePosition ++; while(playQueuePosition < playQueue.length && !playQueue[playQueuePosition].video){ playQueuePosition ++; } if(playQueuePosition < playQueue.length){ playQueueUpdate(); }else if(vp.getReplayCurrent()){ playQueuePosition = 0; playQueueUpdate(); vp.seekTo(playShouldStart); vp.playVideo(); }else{ vp.pauseVideo(); playShouldStop = null; playShouldStart = null; } }; function loadMoreVideos(playerId, vp, start, finish, callback){ var playlistInfo = playlists[playerId-1]; if(playlistInfo.loading >= finish) return; playlistInfo.loading = finish; $.ajax({ url: '/api/upge/cheetah-photo-search/query_videos2', dataType: 'json', data: { query: playlistInfo.query, orderby: playlistInfo.orderby, start: start, count: finish-start }, success: function(response){ var pl = vp.getPlaylist().slice(0); pl.push.apply(pl, response); vp.setPlaylist(pl); callback(); } }); } if(parts.length == 1 && matchDash(parts[0])){ var pl = vp.getActualPlaylist(); var vids = parseDash(parts[0]); parts = []; for(var i = 0; i < vids.length; i++){ playQueue.push({ 'video': pl[vids[i]-1], 'start': 0, 'stop': null }) } if(vids.length){ if(vids[vids.length-1]-1>=pl.length){ loadMoreVideos(playerId, vp, pl.length, vids[vids.length-1], function(){ if(fsonload){ activateTabArea(tab, parts[0]+'/fs'); }else{ activateTabArea(tab, parts[0]); } var pls = vp.getPlaylist(); vp.playFromPlaylist(pls[pls.length-1]); vp.playVideo(); scrollToPlaylistPosition(vp); }); return true; } } if(playQueue){ playQueueUpdate(); vp.playVideo(); parsed = true; playShouldStart = 0; } } if(previewMode){ var vids = []; var dur = 0; var pl = vp.getActualPlaylist(); area = parts[0]; if(parts.length == 1 && matchPosition(parts[0])){ vids = parseDash('1-'+pl.length); dur = parsePosition(parts[0]); parts = []; }else if(parts.length == 1 && matchDash(parts[0])){ vids = parseDash(parts[0]); dur = parsePosition("999:59"); parts = []; } if(parts.length == 2 && matchDash(parts[0]) && matchPosition(parts[1])){ vids = parseDash(parts[0]); dur = parsePosition(parts[1]); parts = []; } for(var i = 0; i < vids.length; i++){ playQueue.push({ 'video': pl[vids[i]-1], 'start': 0, 'stop': dur }) } if(playQueue){ playQueueUpdate(); vp.playVideo(); parsed = true; } } if(parts.length>1){ for(var i = 0; i < parts.length; i++){ var sel = findMatchingVideo(vp, parts[i]); if(sel){ playQueue.push({ 'video': sel, 'start': 0, 'stop': null }) } } if(playQueue){ playQueueUpdate(); vp.playVideo(); parsed = true; } }else if(area){ var sel = findMatchingVideo(vp, area); if(sel){ vp.playFromPlaylist(sel); playShouldStart = 0; parsed = true; } } if(fsonload || replayMode){ playShouldStart = 0; } if(document.location.search.match('at=|queue=')){ var opts = document.location.search.replace(/^\?/,'').split(/&/g); for(var o in opts){ if(opts[o].match(/^at=(\d+:)?(\d+:)?\d+$/)){ playShouldStart = parsePosition(opts[o].substr(3)) } if(opts[o].match(/^queue=/)){ playQueue = parseList(opts[o].substr(6)); if(playQueue){ playQueuePosition = 0; playQueueUpdate(); } } } } if(matchPosition(parts[1])){ playShouldStart = parsePosition(parts[1]); parsed = true; } if(matchAction(parts[1])){ var action = parseAction(vp, area+'/'+parts[1]); playShouldStart = action.start; playShouldStop = action.stop; parsed = true; } if(playShouldStart !== null && !playQueue.length){ playQueue.push({ video: vp.getCurrentVideo(), start: playShouldStart, stop: playShouldStop }); } if(playShouldStart != null){ setInterval(function(){ if(playShouldStop && vp.currentPlayer && vp.currentPlayer.getCurrentTime() > playShouldStop){ playShouldStop = null; if(vp.getCurrentVideo() == playQueue[playQueuePosition].video){ playQueueAdvancePosition(); }else{ playShouldStart = null; } } }, 500); vp.playerContainer.bind('videoplayer.player.statechange', function(e, state){ if(state == 'ended'){ // advance to the next video playQueueAdvancePosition(); } }); vp.playerContainer.bind('videoplayer.player.readychange', function(e, state){ if(state){ updateVolumeState(); if(playShouldStart !== null){ vp.seekTo(playShouldStart); playShouldStart = null; }else{ playShouldStop = null; // someone started other video, stop playing from playQueue } } if(fsonload) { triggerFullscreen(playerId); fsonload = false; } }); } } else if(tab.match(/^wiki\d+$/)){ if(firstTimeActivate){ load_wiki($('#'+tab), function(){ if(area){ var areaNode = $('#'+area); if(areaNode.length>0){ $('html, body').scrollTop(areaNode.offset().top + 10); return true; } } }); } } return parsed; }) window.activateTab = ensure(function(tab, area){ window.activeArea = null; if(tab == 'import_videos'){ if(area){ import_videos(area); }else{ start_import(); } return true; } if(tab == 'chat'){ update_chat_position($('.chat').eq(0)); window.activeArea = 'chat'; jQuery('.tabtrigger').offscreentabs('activateTab', 'chat'); return true; } if(tab in rev_names){ tab = rev_names[tab]; } if(tab.match(':')){ return false; } var sup = $('ul li a[id=#'+tab+']'); if(sup && sup.length>0){ window.activeArea = area; sup.first().click(); if(!window.activateTabArea(tab, area)){ window.activeArea = null; } window.activeArea = null; return true; }else{ var have_tabs = $('#playlist_menu li').length; if(tab.match(/^playlists?\d+$/)){ var to_add = +tab.substring(8).replace(/^s/,'')-have_tabs; if(to_add>0 && have_tabs){ add_more_videos(to_add); return true; } } } return false; }); window.currentPath = ensure(function(){ return window.lastHistory.replace(basepath, '').split('?')[0]; }); window.main_tab = window.main_tab || 'videos'; window.addHistory = ensure(function(path){ if(window.console && console.log) console.log("Adding to history: "+path); if(window.history && history.replaceState && document.location.hostname.match(/^(youtube\.)?(\w{2,3}\.)?wn\.com$/)){ if(path == main_tab || path == main_tab+'/' || path == '' || path == '/') { path = basepath; } else if( path.match('^'+main_tab+'/') ){ path = basepath + '/' + path.replace(main_tab+'/', '').replace('--','/'); } else { path = basepath + '/' + path.replace('--','/'); } if(document.location.search){ path += document.location.search; } if(window.lastHistory) { history.pushState(null, null, path); } else if(window.lastHistory != path){ history.replaceState(null, null, path); window.lastHistory = path; } } else{ path = path.replace('--','/'); if(path == main_tab || path == main_tab+'/' || path == '' || path == '/') { path = ''; } if(window.lastHistory != '/'+path){ window.location.hash = path? '/'+path : ''; window.lastHistory = '/'+path; } } }); $('.tabtrigger li a').live('click', ensure(function() { var tab = $(this).attr('id'); if(tab.substring(0,1) == '#'){ var name = tab.substring(1); if(name in menu_names){ name = menu_names[name][0]; } realTab = rev_names[name]; $('#'+realTab).show(); if(window.console && console.log) console.log("Triggering tab: "+name+(window.activeArea?" activeArea="+window.activeArea:'')); var path = name; if(window.activeArea){ path = path + '/' + window.activeArea; } if(tab.match(/#playlist\d+/) || tab.match(/#details\d+/)){ $('.multiple-playlists').show(); $('.related_playlist').show(); $('.longest_videos_playlist').show(); }else { $('.multiple-playlists').hide(); $('.related_playlist').hide(); $('.longest_videos_playlist').hide(); } // start the related script only when the tab is on screen showing if (tab.match(/related_sites/)) { if (mc) { mc.startCredits(); } } window.activeTab = realTab; addHistory(path); setTimeout(ensure(function(){ if(tab.match(/language--/)){ $('.tabtrigger').offscreentabs('activateTab', 'language'); } if(tab.match(/weather/)) { $('.tabtrigger').offscreentabs('activateTab', 'weather'); loadContinent(); } updateMenus(tab); updateHeight(); }), 10); } return false; })); }); -->
The direct sum is an operation from abstract algebra, a branch of mathematics. As an example, consider the direct sum , where is the set of real numbers. is the Cartesian plane, the xy-plane from elementary algebra. In general, the direct sum of two objects is another object of the same type, so the direct sum of two geometric objects is a geometric object and the direct sum of two sets is a set.
To see how direct sum is used in abstract algebra, consider a more elementary structure in abstract algebra, the abelian group. The direct sum of two abelian groups and is another abelian group consisting of the ordered pairs where and . To add ordered pairs, we define the sum to be ; in other words addition is defined coordinate-wise. A similar process can be used to form the direct sum of any two algebraic structures, such as rings, modules, and vector spaces.
We can also form direct sums with any number of summands, for example , provided and are the same kinds of algebraic structures, that is, all groups or all rings or all vector spaces.
Direct may refer to:
Direct is a 1981 live in-studio album by Tower of Power. It was their only album for the direct-to-disc record label Sheffield Lab. It also marked the return of original guitarist Willie James Fulton, not heard from since 1972's Bump City, and the last album to feature saxophonist Lenny Pickett. Mark Sanders plays drums on this album. Between this album and their 1987 comeback album Power they would record the sessions that later became the Dinosaur Tracks CD. Besides that, save for the original single release of "Simple As That" (from the same Dinosaur Tracks sessions), this would be their last new release until 1987. It contained mostly songs from their previous albums but included new stuff as well.
Direct would later be released with alternate takes on CD as Direct PLUS in 1997.
Side One
Direct is the title of The 77s' second EP, released in 2002 on the band's own Fools of the World label.
University of Oxford mathematician Dr Tom Crawford explains the concept of the direct sum of vector subspaces with several fully-worked examples. Check out ProPrep with a 30-day free trial to see how it can help you to improve your performance in STEM-based subjects: https://www.proprep.uk/info/TOM-Crawford Links to the other videos mentioned: Subspace Test - https://youtu.be/3_MxBlWQsgs Dimension Formula: https://youtu.be/lGo9E5dZ4f8 Test your understanding of the content covered in the video with some practice exercises courtesy of ProPrep. You can download the workbooks and solutions for free here: https://api.proprep.com/course/downloadbook?file=Proprep%20-%20Linear%20Algebra%20-%20General%20Vector%20Spaces%20-%20workbook.pdf You can also find several video lectures from ProPrep e...
Have you ever wondered how to sum two mathematical objects in an elegant way? Then this video is for you! In this video, I define the sum of two vector spaces and show something neat: If you add two bases together, you get a basis for the direct sum! Finally, I generalize this notion to direct sums of n vector spaces. Enjoy! Check out my Vector Space Playlist: https://www.youtube.com/playlist?list=PLJb1qAQIrmmDhpRvkRXUMNGfV3LVZwL_w Subscribe to my channel: youtube.com/c/drpeyam
Descrição
Descrição
⭐Support the channel⭐ Patreon: https://www.patreon.com/michaelpennmath Merch: https://teespring.com/stores/michael-penn-math My amazon shop: https://www.amazon.com/shop/michaelpenn ⭐my other channels⭐ Main Channel: https://www.youtube.com/michaelpennmath non-math podcast: https://www.youtube.com/channel/UCIl9oLq0igolmqLYipO0OBQ ⭐My Links⭐ Personal Website: http://www.michael-penn.net Instagram: https://www.instagram.com/melp2718/ Randolph College Math: http://www.randolphcollege.edu/mathematics/ Research Gate profile: https://www.researchgate.net/profile/Michael_Penn5 Google Scholar profile: https://scholar.google.com/citations?user=W5wkSxcAAAAJ&hl=en
Algebra 1M - international Course no. 104016 Dr. Aviv Censor Technion - International school of engineering
Direct Sum definition In this video, I define the notion of direct sum of n subspaces and show what it has to do with eigenvectors. Direct sum of two subspaces: https://youtu.be/GjbMddz0Qxs Check out my Diagonalization playlist: https://www.youtube.com/playlist?list=PLJb1qAQIrmmCSovHY6cXzPMNSuWOwd9wB Subscribe to my channel: https://www.youtube.com/c/drpeyam
This lecture explains the direct sum of the subspaces and its illustrative examples. Other videos @DrHarishGarg Row reduced Echelon form: https://youtu.be/yCTfbsf60NA Rank of the matrix: https://youtu.be/Hd34nBwcBG8 Vector Subspaces: https://youtu.be/9uN7CwDqvZQ Intersection & Union of the subspaces: https://youtu.be/UtUvk1k5YIU Linear Span and Combination: https://youtu.be/dT0DHyE2Fz4 Direct Sum of Subspace: https://youtu.be/W1RTq9ipXI8
🔴 Regardez FRANCE 24 en français en direct gratuitement et en streaming sur YouTube. Suivez toutes les infos du monde en continu. 🔔 Abonnez-vous à notre chaîne sur YouTube : https://f24.my/YTfr 💬 Commentez l'actualité du jour sur le chat #DirectF24 ! Partagez vos commentaires et réactions sur nos journaux, émissions, interviews et reportages. Merci toutefois de faire de cet espace un lieu d'échange et de respect mutuel. France 24 n'est pas responsable des commentaires laissés par les visiteurs. France 24 LIVE in English: https://f24.my/YTliveEN France 24 EN VIVO en Español: https://f24.my/YTliveES فرانس 24 البث المباشر https://f24.my/YTliveAR #France24 #Direct #TV 🔔 Abonnez-vous à notre chaîne sur YouTube : https://f24.my/YTfr 🔴 En DIRECT - Suivez FRANCE 24 ici : ht...
🛑 [DIRECT] Bloc Matinale Avec Asma Khadija, Yacine Seck,Mame Mbaye Ndiaye Revue des titres DU MERCREDI 22 JANVIER 2025📍Vous pouvez réagir en direct # #Kebetu #ZikFm #SenTv #Dmedia #Team221 #diaspora #senegal @TFMTeleFutursMedias @WalfadjriTV @2stvsenegal
Pour plus de vidéos, abonne- z-vous sur notre chaine http://www.youtube.com/tfm https://www.facebook.com/T https://www.ins tagram.com
Suivez toute l'actualité sur http://www.francetvinfo.fr Disponible 24h/24 et 7j/7 sur le canal 27 de la TNT, sur les box, et sur tous les écrans numériques, la chaîne TV d’information en continu franceinfo propose des émissions diffusées simultanément à la radio sur certaines tranches horaires, des JT à l’heure et à la demi-heure (6 sont sous-titrés et 2 sont en langue des signes chaque jour), des rappels des titres, des modules d’analyse et d’approfondissement ainsi que des duplex réunissant toutes les rédactions partenaires. Toute l'info en direct et les meilleures vidéos en streaming et Replay. Actualité France et Monde, Interviews, Documentaires et Analyses.
Urgent 🛑 Entretien exclusif avec Aïssatou Diop Fall !
🔴[DIRECT] L'invité de #MNF: Ngouda FALL KANE sur la CENTIF; le parquet financier et... #senegal #7TV #Kebetu
#kebetu #senegal #team221 🔴Direct: La décision est tombée pour Farba Ngom, Aïssata Talla Sall et cie face à la presse
Thématiques Grandes Gueules : 1- Abou Diallo placé en garde à vue pour incitation à la discrimination Ethnique : du Projet aux procés ,la justice est-elle utilisé à des fins politiques ? 2-Un Constat largement partagé : "Rewmi Dokhoul " le Président admet que son régime est en face d'une Marge de Manœuvre budgétaire et financière Nulles.Quelles solutions pour une sortie de crise?📍Vous pouvez réagir en direct # #Kebetu #ZikFm #SenTv #Dmedia #Team221 #Diasporaa @Senegal7tv @WalfadjriTV
Investiture Trump en direct : le moment où Donald Trump a prêté serment comme 47e président des Etats-Unis d'Amérique au Capitole aux côté de son nouveau vice-président JD Vance et devant de nombreux invités notamment d'anciens présidents. ABONNEZ-VOUS pour plus de vidéos : http://bit.ly/radioE1 LE DIRECT : http://www.europe1.fr/direct-video Retrouvez-nous sur : | Notre site : http://www.europe1.fr | Facebook : https://www.facebook.com/Europe1 | Twitter : https://twitter.com/europe1 | Instagram : https://www.instagram.com/europe1/ #europe1 #cnews
Vivez en direct le match de Ligue des Champions entre le Benfica Lisbonne et le FC Barcelone. #BENBAR #BenficaBarça #LDC 💲 TU VEUX DEVENIR SPONSOR et avoir un max d'avantages ? 👇clique sur le lien https://www.youtube.com/channel/UCzSSwnU5MohO6vQqmMjKUyw/join 🔹 Mes réseaux sociaux : ✔ Mail Pro : [email protected] ✔ Twitch : https://www.twitch.tv/limmigreparisien ✔ Twitter : https://twitter.com/ImmigreParisien ✔ Insta : https://www.instagram.com/immigreparisien/?hl=fr live ldc,direct ldc,streaming ldc,ldc live,ldc direct,ldc match,ldc en direct,ldc streaming,ucl live,ucl direct,ucl en direct,ucl match,live ucl,match ucl,direct ucl en direct ucl,match en direct aujourd'hui,match ldc live,match ucl live,multipex ldc live,champions league,ligue des champions,ucl,ldc,live champion...
SHEFFIELD LAB 17 PRODUCED BY DOUG SAX, LINCOLN MAYORGA, AND LARRY BROWN
00:00 A1 Baby's Got The Power 03:09 A2 Credit 07:16 A3 Some Days Were Meant For Rain 12:11 A4 Boys Night Out 18:05 B1 Ball And Chain 21:21 B2 Through Lover's Eyes 25:02 B3 Count On Me 28:31 B4 On The One 32:36 B5 Up Against Yourself
Tower Of Power Direct Sheffield Lab CD-17 US, 1981 Collection P. Kritidis 00:00 Fanfare : You Know It Composed by, Arranged by - Greg Adams Written by - Emilio Castillo, Stephen Kupka 02:47 You 're Gonna Need Me Lead Vocals - Willie James Fulton Written by - Albert King 06:55 Squib Cakes Composed by, Arranged by - Chester Thompson (2) Flugelhorn, Soloist - Greg Adams. Saxophone, Soloist - Lenny Pickett 14:49 That's Why I Sing Composed by, Arranged by - Chester Thompson (2) 19:49 What Is Hip Trumpet, Soloist - Greg Adams Written by - Emilio Castillo, Stephen Kupka 24:35 Never Let Go Of Love Arranged by - Greg Adams Saxophone,...
Provided to YouTube by The Orchard Enterprises Squibcakes · Tower of Power Direct Plus! ℗ 1981 Sheffield Lab Inc. Released on: 2007-04-05 Auto-generated by YouTube.
Provided to YouTube by The Orchard Enterprises What Is Hip (previously unreleased) · Tower of Power Direct Plus! ℗ 1981 Sheffield Lab Inc. Released on: 2007-04-05 Auto-generated by YouTube.
Provided to YouTube by The Orchard Enterprises What Is Hip · Tower of Power Direct Plus! ℗ 1981 Sheffield Lab Inc. Released on: 2007-04-05 Auto-generated by YouTube.
For 55 years, Tower of Power has delivered the best in funk and soul music.“We were a Soul band called The Motowns.” recalls Emilio Castillo. “Rocco was the bass player, I was there, and my brother was the drummer. I met “Doc” Kupka back in 1968, and gave him an audition. He came in the band, and we eventually changed our name to the Tower of Power.” The reason for the band name change was that they had a specific goal in mind. East Bay Grease defined their sound and landed them with Warner Brothers. Bump City, their 1972 debut for the label, was a hit on both the Billboard 200 and the R&B Albums chart, and netted them the hits “You’re Still A Young Man” and “Down to The Nightclub.” The 70s were a boom period with radio classics like “So Very Hard to Go” and “What Is Hip?,” and the band ...
Provided to YouTube by The Orchard Enterprises That's Why I Sing · Tower of Power Direct Plus! ℗ 1981 Sheffield Lab Inc. Released on: 2007-04-05 Auto-generated by YouTube.
Provided to YouTube by The Orchard Enterprises Never Let Go Of Love · Tower of Power Direct Plus! ℗ 1981 Sheffield Lab Inc. Released on: 2007-04-05 Auto-generated by YouTube.
Provided to YouTube by The Orchard Enterprises Squibcakes (previously unreleased) · Tower of Power Direct Plus! ℗ 1981 Sheffield Lab Inc. Released on: 2007-04-05 Auto-generated by YouTube.
Provided to YouTube by The Orchard Enterprises You're Gonna Need Me · Tower of Power Direct Plus! ℗ 1981 Sheffield Lab Inc. Released on: 2007-04-05 Auto-generated by YouTube.
Provided to YouTube by The Orchard Enterprises Fanfare/You Know It · Tower of Power Direct Plus! ℗ 1981 Sheffield Lab Inc. Released on: 2007-04-05 Auto-generated by YouTube.
Classic Direct to Disc Recording of Squib Cakes!
Provided to YouTube by The Orchard Enterprises Fanfare/You Know It (previously unreleased) · Tower of Power Direct Plus! ℗ 1981 Sheffield Lab Inc. Released on: 2007-04-05 Auto-generated by YouTube.
The direct sum is an operation from abstract algebra, a branch of mathematics. As an example, consider the direct sum , where is the set of real numbers. is the Cartesian plane, the xy-plane from elementary algebra. In general, the direct sum of two objects is another object of the same type, so the direct sum of two geometric objects is a geometric object and the direct sum of two sets is a set.
To see how direct sum is used in abstract algebra, consider a more elementary structure in abstract algebra, the abelian group. The direct sum of two abelian groups and is another abelian group consisting of the ordered pairs where and . To add ordered pairs, we define the sum to be ; in other words addition is defined coordinate-wise. A similar process can be used to form the direct sum of any two algebraic structures, such as rings, modules, and vector spaces.
We can also form direct sums with any number of summands, for example , provided and are the same kinds of algebraic structures, that is, all groups or all rings or all vector spaces.