Linear Algebra nTuples and Rn











>> YOUR LINK HERE: ___ http://youtube.com/watch?v=Uh0lZ-s2BHQ

In this video, we discuss the idea of R^n, n-tuples and configuration space which is the first part of linear algebra and how it relates space and talks about how to describe it mathematically.","styleRuns":[{"startIndex":0,"length":193,"styleRunExtensions":{"styleRunColorMapExtension":{"colorMap":[{"key":"USER_INTERFACE_THEME_DARK","value":4294967295},{"key":"USER_INTERFACE_THEME_LIGHT","value":4279440147}]}},"fontFamilyName":"Roboto"}]},"headerRuns":[{"startIndex":0,"length":193,"headerMapping":"ATTRIBUTED_STRING_HEADER_MAPPING_UNSPECIFIED"}]}},{"itemSectionRenderer":{"contents":[{"continuationItemRenderer":{"trigger":"CONTINUATION_TRIGGER_ON_ITEM_SHOWN","continuationEndpoint":{"clickTrackingParams":"CNABELsvGAIiEwjt1uq03oSMAxWZRnoFHdkuMMA=","commandMetadata":{"webCommandMetadata":{"sendPost":true,"apiUrl":"/youtubei/v1/next"}},"continuationCommand":{"token":"Eg0SC1VoMGxaLXMyQkhRGAYyJSIRIgtVaDBsWi1zMkJIUTAAeAJCEGNvbW1lbnRzLXNlY3Rpb24%3D","request":"CONTINUATION_REQUEST_TYPE_WATCH_NEXT"}}}}],"trackingParams":"CNABELsvGAIiEwjt1uq03oSMAxWZRnoFHdkuMMA=","sectionIdentifier":"comment-item-section","targetId":"comments-section"}}],"trackingParams":"CM8BELovIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA"}},"secondaryResults":{"secondaryResults":{"results":[{"compactVideoRenderer":{"videoId":"_iSgauACGdY","thumbnail":{"thumbnails":[{"url":"https://i.ytimg.com/vi/_iSgauACGdY/hqdefault.jpg?sqp=-oaymwEiCKgBEF5IWvKriqkDFQgBFQAAAAAYASUAAMhCPQCAokN4AQ== rs=AOn4CLD9oXuJyUPVwURSRjqOMh-l5RJHLQ","width":168,"height":94},{"url":"https://i.ytimg.com/vi/_iSgauACGdY/hqdefault.jpg?sqp=-oaymwEjCNACELwBSFryq4qpAxUIARUAAAAAGAElAADIQj0AgKJDeAE= rs=AOn4CLDavRVlNIbtgyiTOtm-70BMlYrSPQ","width":336,"height":188}]},"title":{"accessibility":{"accessibilityData":{"label":"Linear Algebra: Points, Arrows and Vectors by Math for Thought 4,446 views 4 years ago 38 minutes"}},"simpleText":"Linear Algebra: Points, Arrows and Vectors"},"longBylineText":{"runs":[{"text":"Math for Thought","navigationEndpoint":{"clickTrackingParams":"CMkBEKQwGAAiEwjt1uq03oSMAxWZRnoFHdkuMMAyB3JlbGF0ZWQ=","commandMetadata":{"webCommandMetadata":{"url":"/@MathforThought","webPageType":"WEB_PAGE_TYPE_CHANNEL","rootVe":3611,"apiUrl":"/youtubei/v1/browse"}},"browseEndpoint":{"browseId":"UCeMTK2EkNHiTBw3LroYjoXQ","canonicalBaseUrl":"/@MathforThought"}}}]},"publishedTimeText":{"simpleText":"4 years ago"},"viewCountText":{"simpleText":"4,446 views"},"lengthText":{"accessibility":{"accessibilityData":{"label":"38 minutes, 54 seconds"}},"simpleText":"38:54"},"navigationEndpoint":{"clickTrackingParams":"CMkBEKQwGAAiEwjt1uq03oSMAxWZRnoFHdkuMMAyB3JlbGF0ZWRI9IjY2f6syY5SmgEFCAEQ-B0=","commandMetadata":{"webCommandMetadata":{"url":"/watch?v=_iSgauACGdY","webPageType":"WEB_PAGE_TYPE_WATCH","rootVe":3832}},"watchEndpoint":{"videoId":"_iSgauACGdY","nofollow":true,"watchEndpointSupportedOnesieConfig":{"html5PlaybackOnesieConfig":{"commonConfig":{"url":"https://rr1---sn-uxaxiv0nxx5q-nv4l.googlevideo.com/initplayback?source=youtube oeis=1 c=WEB oad=3200 ovd=3200 oaad=11000 oavd=11000 ocs=700 oewis=1 oputc=1 ofpcc=1 msp=1 odepv=1 onvi=1 id=fe24a06ae00219d6 ip=2a02%3A27aa%3A0%3A0%3A0%3A0%3A0%3Ad49 initcwndbps=1396250 mt=1741788795 oweuc="}}}}},"shortBylineText":{"runs":[{"text":"Math for Thought","navigationEndpoint":{"clickTrackingParams":"CMkBEKQwGAAiEwjt1uq03oSMAxWZRnoFHdkuMMAyB3JlbGF0ZWQ=","commandMetadata":{"webCommandMetadata":{"url":"/@MathforThought","webPageType":"WEB_PAGE_TYPE_CHANNEL","rootVe":3611,"apiUrl":"/youtubei/v1/browse"}},"browseEndpoint":{"browseId":"UCeMTK2EkNHiTBw3LroYjoXQ","canonicalBaseUrl":"/@MathforThought"}}}]},"channelThumbnail":{"thumbnails":[{"url":"https://yt3.ggpht.com/ytc/AIdro_mbTAP7kl9p89r3hItqfV4epIqTuVd30aLd55Khhg4rGg=s68-c-k-c0x00ffffff-no-rj","width":68,"height":68}]},"trackingParams":"CMkBEKQwGAAiEwjt1uq03oSMAxWZRnoFHdkuMMBA1rOIgK6NqJL-AQ==","shortViewCountText":{"accessibility":{"accessibilityData":{"label":"4.4K views"}},"simpleText":"4.4K views"},"menu":{"menuRenderer":{"items":[{"menuServiceItemRenderer":{"text":{"runs":[{"text":"Add to queue"}]},"icon":{"iconType":"ADD_TO_QUEUE_TAIL"},"serviceEndpoint":{"clickTrackingParams":"CM0BEP6YBBgFIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA","commandMetadata":{"webCommandMetadata":{"sendPost":true}},"signalServiceEndpoint":{"signal":"CLIENT_SIGNAL","actions":[{"clickTrackingParams":"CM0BEP6YBBgFIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA","addToPlaylistCommand":{"openMiniplayer":false,"openListPanel":true,"videoId":"_iSgauACGdY","listType":"PLAYLIST_EDIT_LIST_TYPE_QUEUE","onCreateListCommand":{"clickTrackingParams":"CM0BEP6YBBgFIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA","commandMetadata":{"webCommandMetadata":{"sendPost":true,"apiUrl":"/youtubei/v1/playlist/create"}},"createPlaylistServiceEndpoint":{"videoIds":["_iSgauACGdY"],"params":"CAQ%3D"}},"videoIds":["_iSgauACGdY"]}},{"clickTrackingParams":"CM0BEP6YBBgFIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA","openPopupAction":{"popup":{"notificationActionRenderer":{"responseText":{"simpleText":"Added to queue"},"trackingParams":"CM4BELlqIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA"}},"popupType":"TOAST"}}]}},"trackingParams":"CM0BEP6YBBgFIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA"}},{"menuServiceItemRenderer":{"text":{"runs":[{"text":"Share"}]},"icon":{"iconType":"SHARE"},"serviceEndpoint":{"clickTrackingParams":"CMkBEKQwGAAiEwjt1uq03oSMAxWZRnoFHdkuMMA=","commandMetadata":{"webCommandMetadata":{"sendPost":true,"apiUrl":"/youtubei/v1/share/get_share_panel"}},"shareEntityServiceEndpoint":{"serializedShareEntity":"CgtfaVNnYXVBQ0dkWQ%3D%3D","commands":[{"clickTrackingParams":"CMkBEKQwGAAiEwjt1uq03oSMAxWZRnoFHdkuMMA=","openPopupAction":{"popup":{"unifiedSharePanelRenderer":{"trackingParams":"CMwBEI5iIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA","showLoadingSpinner":true}},"popupType":"DIALOG","beReused":true}}]}},"trackingParams":"CMkBEKQwGAAiEwjt1uq03oSMAxWZRnoFHdkuMMA=","hasSeparator":true}}],"trackingParams":"CMkBEKQwGAAiEwjt1uq03oSMAxWZRnoFHdkuMMA=","accessibility":{"accessibilityData":{"label":"Action menu"}},"targetId":"watch-related-menu-button"}},"thumbnailOverlays":[{"thumbnailOverlayTimeStatusRenderer":{"text":{"accessibility":{"accessibilityData":{"label":"38 minutes, 54 seconds"}},"simpleText":"38:54"},"style":"DEFAULT"}},{"thumbnailOverlayToggleButtonRenderer":{"isToggled":false,"untoggledIcon":{"iconType":"WATCH_LATER"},"toggledIcon":{"iconType":"CHECK"},"untoggledTooltip":"Watch later","toggledTooltip":"Added","untoggledServiceEndpoint":{"clickTrackingParams":"CMsBEPnnAxgBIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA","commandMetadata":{"webCommandMetadata":{"sendPost":true,"apiUrl":"/youtubei/v1/browse/edit_playlist"}},"playlistEditEndpoint":{"playlistId":"WL","actions":[{"addedVideoId":"_iSgauACGdY","action":"ACTION_ADD_VIDEO"}]}},"toggledServiceEndpoint":{"clickTrackingParams":"CMsBEPnnAxgBIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA","commandMetadata":{"webCommandMetadata":{"sendPost":true,"apiUrl":"/youtubei/v1/browse/edit_playlist"}},"playlistEditEndpoint":{"playlistId":"WL","actions":[{"action":"ACTION_REMOVE_VIDEO_BY_VIDEO_ID","removedVideoId":"_iSgauACGdY"}]}},"untoggledAccessibility":{"accessibilityData":{"label":"Watch later"}},"toggledAccessibility":{"accessibilityData":{"label":"Added"}},"trackingParams":"CMsBEPnnAxgBIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA"}},{"thumbnailOverlayToggleButtonRenderer":{"untoggledIcon":{"iconType":"ADD_TO_QUEUE_TAIL"},"toggledIcon":{"iconType":"PLAYLIST_ADD_CHECK"},"untoggledTooltip":"Add to queue","toggledTooltip":"Added","untoggledServiceEndpoint":{"clickTrackingParams":"CMoBEMfsBBgCIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA","commandMetadata":{"webCommandMetadata":{"sendPost":true}},"signalServiceEndpoint":{"signal":"CLIENT_SIGNAL","actions":[{"clickTrackingParams":"CMoBEMfsBBgCIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA","addToPlaylistCommand":{"openMiniplayer":false,"openListPanel":true,"videoId":"_iSgauACGdY","listType":"PLAYLIST_EDIT_LIST_TYPE_QUEUE","onCreateListCommand":{"clickTrackingParams":"CMoBEMfsBBgCIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA","commandMetadata":{"webCommandMetadata":{"sendPost":true,"apiUrl":"/youtubei/v1/playlist/create"}},"createPlaylistServiceEndpoint":{"videoIds":["_iSgauACGdY"],"params":"CAQ%3D"}},"videoIds":["_iSgauACGdY"]}}]}},"untoggledAccessibility":{"accessibilityData":{"label":"Add to queue"}},"toggledAccessibility":{"accessibilityData":{"label":"Added"}},"trackingParams":"CMoBEMfsBBgCIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA"}},{"thumbnailOverlayNowPlayingRenderer":{"text":{"runs":[{"text":"Now playing"}]}}}],"accessibility":{"accessibilityData":{"label":"Linear Algebra: Points, Arrows and Vectors - 38 minutes - Go to channel - Math for Thought - 4.4K views - 4 years ago - play video"}}}},{"compactVideoRenderer":{"videoId":"TgKwz5Ikpc8","thumbnail":{"thumbnails":[{"url":"https://i.ytimg.com/vi/TgKwz5Ikpc8/hqdefault.jpg?sqp=-oaymwEiCKgBEF5IWvKriqkDFQgBFQAAAAAYASUAAMhCPQCAokN4AQ== rs=AOn4CLCRQ7HfVhgcCdgH1wixuSbZCJQ5fA","width":168,"height":94},{"url":"https://i.ytimg.com/vi/TgKwz5Ikpc8/hqdefault.jpg?sqp=-oaymwEjCNACELwBSFryq4qpAxUIARUAAAAAGAElAADIQj0AgKJDeAE= rs=AOn4CLBjIWhJAxO4QboXBG3viLz6zNT6NQ","width":336,"height":188}]},"title":{"accessibility":{"accessibilityData":{"label":"Abstract vector spaces | Chapter 16, Essence of linear algebra by 3Blue1Brown 1,518,350 views 8 years ago 16 minutes"}},"simpleText":"Abstract vector spaces | Chapter 16, Essence of linear algebra"},"longBylineText":{"runs":[{"text":"3Blue1Brown","navigationEndpoint":{"clickTrackingParams":"CMMBEKQwGAEiEwjt1uq03oSMAxWZRnoFHdkuMMAyB3JlbGF0ZWQ=","commandMetadata":{"webCommandMetadata":{"url":"/@3blue1brown","webPageType":"WEB_PAGE_TYPE_CHANNEL","rootVe":3611,"apiUrl":"/youtubei/v1/browse"}},"browseEndpoint":{"browseId":"UCYO_jab_esuFRV4b17AJtAw","canonicalBaseUrl":"/@3blue1brown"}}}]},"publishedTimeText":{"simpleText":"8 years ago"},"viewCountText":{"simpleText":"1,518,350 views"},"lengthText":{"accessibility":{"accessibilityData":{"label":"16 minutes, 46 seconds"}},"simpleText":"16:46"},"navigationEndpoint":{"clickTrackingParams":"CMMBEKQwGAEiEwjt1uq03oSMAxWZRnoFHdkuMMAyB3JlbGF0ZWRI9IjY2f6syY5SmgEFCAEQ-B0=","commandMetadata":{"webCommandMetadata":{"url":"/watch?v=TgKwz5Ikpc8","webPageType":"WEB_PAGE_TYPE_WATCH","rootVe":3832}},"watchEndpoint":{"videoId":"TgKwz5Ikpc8","nofollow":true,"watchEndpointSupportedOnesieConfig":{"html5PlaybackOnesieConfig":{"commonConfig":{"url":"https://rr3---sn-nvm-cxbz.googlevideo.com/initplayback?source=youtube oeis=1 c=WEB oad=3200 ovd=3200 oaad=11000 oavd=11000 ocs=700 oewis=1 oputc=1 ofpcc=1 msp=1 odepv=1 onvi=1 id=4e02b0cf9224a5cf ip=2a02%3A27aa%3A0%3A0%3A0%3A0%3A0%3Ad49 initcwndbps=647500 mt=1741788795 oweuc="}}}}},"shortBylineText":{"runs":[{"text":"3Blue1Brown","navigationEndpoint":{"clickTrackingParams":"CMMBEKQwGAEiEwjt1uq03oSMAxWZRnoFHdkuMMAyB3JlbGF0ZWQ=","commandMetadata":{"webCommandMetadata":{"url":"/@3blue1brown","webPageType":"WEB_PAGE_TYPE_CHANNEL","rootVe":3611,"apiUrl":"/youtubei/v1/browse"}},"browseEndpoint":{"browseId":"UCYO_jab_esuFRV4b17AJtAw","canonicalBaseUrl":"/@3blue1brown"}}}]},"channelThumbnail":{"thumbnails":[{"url":"https://yt3.ggpht.com/ytc/AIdro_nFzZFPLxPZRHcE3SSwzdrbuWqfoWYwLAu0_2iO6blQYAU=s68-c-k-c0x00ffffff-no-rj","width":68,"height":68}]},"ownerBadges":[{"metadataBadgeRenderer":{"icon":{"iconType":"CHECK_CIRCLE_THICK"},"style":"BADGE_STYLE_TYPE_VERIFIED","tooltip":"Verified","trackingParams":"CMMBEKQwGAEiEwjt1uq03oSMAxWZRnoFHdkuMMA=","accessibilityData":{"label":"Verified"}}}],"trackingParams":"CMMBEKQwGAEiEwjt1uq03oSMAxWZRnoFHdkuMMBAz8uSkfmZrIFO","shortViewCountText":{"accessibility":{"accessibilityData":{"label":"1.5 million views"}},"simpleText":"1.5M views"},"menu":{"menuRenderer":{"items":[{"menuServiceItemRenderer":{"text":{"runs":[{"text":"Add to queue"}]},"icon":{"iconType":"ADD_TO_QUEUE_TAIL"},"serviceEndpoint":{"clickTrackingParams":"CMcBEP6YBBgGIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA","commandMetadata":{"webCommandMetadata":{"sendPost":true}},"signalServiceEndpoint":{"signal":"CLIENT_SIGNAL","actions":[{"clickTrackingParams":"CMcBEP6YBBgGIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA","addToPlaylistCommand":{"openMiniplayer":false,"openListPanel":true,"videoId":"TgKwz5Ikpc8","listType":"PLAYLIST_EDIT_LIST_TYPE_QUEUE","onCreateListCommand":{"clickTrackingParams":"CMcBEP6YBBgGIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA","commandMetadata":{"webCommandMetadata":{"sendPost":true,"apiUrl":"/youtubei/v1/playlist/create"}},"createPlaylistServiceEndpoint":{"videoIds":["TgKwz5Ikpc8"],"params":"CAQ%3D"}},"videoIds":["TgKwz5Ikpc8"]}},{"clickTrackingParams":"CMcBEP6YBBgGIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA","openPopupAction":{"popup":{"notificationActionRenderer":{"responseText":{"simpleText":"Added to queue"},"trackingParams":"CMgBELlqIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA"}},"popupType":"TOAST"}}]}},"trackingParams":"CMcBEP6YBBgGIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA"}},{"menuServiceItemRenderer":{"text":{"runs":[{"text":"Share"}]},"icon":{"iconType":"SHARE"},"serviceEndpoint":{"clickTrackingParams":"CMMBEKQwGAEiEwjt1uq03oSMAxWZRnoFHdkuMMA=","commandMetadata":{"webCommandMetadata":{"sendPost":true,"apiUrl":"/youtubei/v1/share/get_share_panel"}},"shareEntityServiceEndpoint":{"serializedShareEntity":"CgtUZ0t3ejVJa3BjOA%3D%3D","commands":[{"clickTrackingParams":"CMMBEKQwGAEiEwjt1uq03oSMAxWZRnoFHdkuMMA=","openPopupAction":{"popup":{"unifiedSharePanelRenderer":{"trackingParams":"CMYBEI5iIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA","showLoadingSpinner":true}},"popupType":"DIALOG","beReused":true}}]}},"trackingParams":"CMMBEKQwGAEiEwjt1uq03oSMAxWZRnoFHdkuMMA=","hasSeparator":true}}],"trackingParams":"CMMBEKQwGAEiEwjt1uq03oSMAxWZRnoFHdkuMMA=","accessibility":{"accessibilityData":{"label":"Action menu"}}}},"thumbnailOverlays":[{"thumbnailOverlayTimeStatusRenderer":{"text":{"accessibility":{"accessibilityData":{"label":"16 minutes, 46 seconds"}},"simpleText":"16:46"},"style":"DEFAULT"}},{"thumbnailOverlayToggleButtonRenderer":{"isToggled":false,"untoggledIcon":{"iconType":"WATCH_LATER"},"toggledIcon":{"iconType":"CHECK"},"untoggledTooltip":"Watch later","toggledTooltip":"Added","untoggledServiceEndpoint":{"clickTrackingParams":"CMUBEPnnAxgCIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA","commandMetadata":{"webCommandMetadata":{"sendPost":true,"apiUrl":"/youtubei/v1/browse/edit_playlist"}},"playlistEditEndpoint":{"playlistId":"WL","actions":[{"addedVideoId":"TgKwz5Ikpc8","action":"ACTION_ADD_VIDEO"}]}},"toggledServiceEndpoint":{"clickTrackingParams":"CMUBEPnnAxgCIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA","commandMetadata":{"webCommandMetadata":{"sendPost":true,"apiUrl":"/youtubei/v1/browse/edit_playlist"}},"playlistEditEndpoint":{"playlistId":"WL","actions":[{"action":"ACTION_REMOVE_VIDEO_BY_VIDEO_ID","removedVideoId":"TgKwz5Ikpc8"}]}},"untoggledAccessibility":{"accessibilityData":{"label":"Watch later"}},"toggledAccessibility":{"accessibilityData":{"label":"Added"}},"trackingParams":"CMUBEPnnAxgCIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA"}},{"thumbnailOverlayToggleButtonRenderer":{"untoggledIcon":{"iconType":"ADD_TO_QUEUE_TAIL"},"toggledIcon":{"iconType":"PLAYLIST_ADD_CHECK"},"untoggledTooltip":"Add to queue","toggledTooltip":"Added","untoggledServiceEndpoint":{"clickTrackingParams":"CMQBEMfsBBgDIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA","commandMetadata":{"webCommandMetadata":{"sendPost":true}},"signalServiceEndpoint":{"signal":"CLIENT_SIGNAL","actions":[{"clickTrackingParams":"CMQBEMfsBBgDIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA","addToPlaylistCommand":{"openMiniplayer":false,"openListPanel":true,"videoId":"TgKwz5Ikpc8","listType":"PLAYLIST_EDIT_LIST_TYPE_QUEUE","onCreateListCommand":{"clickTrackingParams":"CMQBEMfsBBgDIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA","commandMetadata":{"webCommandMetadata":{"sendPost":true,"apiUrl":"/youtubei/v1/playlist/create"}},"createPlaylistServiceEndpoint":{"videoIds":["TgKwz5Ikpc8"],"params":"CAQ%3D"}},"videoIds":["TgKwz5Ikpc8"]}}]}},"untoggledAccessibility":{"accessibilityData":{"label":"Add to queue"}},"toggledAccessibility":{"accessibilityData":{"label":"Added"}},"trackingParams":"CMQBEMfsBBgDIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA"}},{"thumbnailOverlayNowPlayingRenderer":{"text":{"runs":[{"text":"Now playing"}]}}}],"accessibility":{"accessibilityData":{"label":"Abstract vector spaces | Chapter 16, Essence of linear algebra - 16 minutes - Go to channel - 3Blue1Brown - 1.5M views - 8 years ago - play video"}}}},{"compactVideoRenderer":{"videoId":"w7apAOKgBYI","thumbnail":{"thumbnails":[{"url":"https://i.ytimg.com/vi/w7apAOKgBYI/hqdefault.jpg?sqp=-oaymwEiCKgBEF5IWvKriqkDFQgBFQAAAAAYASUAAMhCPQCAokN4AQ== rs=AOn4CLBVvduEUFb_at6RHWpKZuJ4G1CNPg","width":168,"height":94},{"url":"https://i.ytimg.com/vi/w7apAOKgBYI/hqdefault.jpg?sqp=-oaymwEjCNACELwBSFryq4qpAxUIARUAAAAAGAElAADIQj0AgKJDeAE= rs=AOn4CLA0gFMUNRtm7AyCKe09AX1m0hrZ6w","width":336,"height":188}]},"title":{"accessibility":{"accessibilityData":{"label":"Lecture 1(B): Sets and n-tuples by Arizona Math Camp 7,614 views 6 years ago 21 minutes"}},"simpleText":"Lecture 1(B): Sets and n-tuples"},"longBylineText":{"runs":[{"text":"Arizona Math Camp","navigationEndpoint":{"clickTrackingParams":"CL0BEKQwGAIiEwjt1uq03oSMAxWZRnoFHdkuMMAyB3JlbGF0ZWQ=","commandMetadata":{"webCommandMetadata":{"url":"/@ArizonaMathCamp","webPageType":"WEB_PAGE_TYPE_CHANNEL","rootVe":3611,"apiUrl":"/youtubei/v1/browse"}},"browseEndpoint":{"browseId":"UCnm29LeklN7kcvHAZ0Fm42g","canonicalBaseUrl":"/@ArizonaMathCamp"}}}]},"publishedTimeText":{"simpleText":"6 years ago"},"viewCountText":{"simpleText":"7,614 views"},"lengthText":{"accessibility":{"accessibilityData":{"label":"21 minutes, 40 seconds"}},"simpleText":"21:40"},"navigationEndpoint":{"clickTrackingParams":"CL0BEKQwGAIiEwjt1uq03oSMAxWZRnoFHdkuMMAyB3JlbGF0ZWRI9IjY2f6syY5SmgEFCAEQ-B0=","commandMetadata":{"webCommandMetadata":{"url":"/watch?v=w7apAOKgBYI","webPageType":"WEB_PAGE_TYPE_WATCH","rootVe":3832}},"watchEndpoint":{"videoId":"w7apAOKgBYI","nofollow":true,"watchEndpointSupportedOnesieConfig":{"html5PlaybackOnesieConfig":{"commonConfig":{"url":"https://rr1---sn-uxaxiv0nxx5q-nv4l.googlevideo.com/initplayback?source=youtube oeis=1 c=WEB oad=3200 ovd=3200 oaad=11000 oavd=11000 ocs=700 oewis=1 oputc=1 ofpcc=1 msp=1 odepv=1 onvi=1 id=c3b6a900e2a00582 ip=2a02%3A27aa%3A0%3A0%3A0%3A0%3A0%3Ad49 initcwndbps=1396250 mt=1741788795 oweuc="}}}}},"shortBylineText":{"runs":[{"text":"Arizona Math Camp","navigationEndpoint":{"clickTrackingParams":"CL0BEKQwGAIiEwjt1uq03oSMAxWZRnoFHdkuMMAyB3JlbGF0ZWQ=","commandMetadata":{"webCommandMetadata":{"url":"/@ArizonaMathCamp","webPageType":"WEB_PAGE_TYPE_CHANNEL","rootVe":3611,"apiUrl":"/youtubei/v1/browse"}},"browseEndpoint":{"browseId":"UCnm29LeklN7kcvHAZ0Fm42g","canonicalBaseUrl":"/@ArizonaMathCamp"}}}]},"channelThumbnail":{"thumbnails":[{"url":"https://yt3.ggpht.com/ytc/AIdro_k25hs58pfKkM2tz825xjC4MzJ9Ol2DppIfnAabHxBWVA=s68-c-k-c0x00ffffff-no-rj","width":68,"height":68}]},"trackingParams":"CL0BEKQwGAIiEwjt1uq03oSMAxWZRnoFHdkuMMBAgouAlY6gqtvDAQ==","shortViewCountText":{"accessibility":{"accessibilityData":{"label":"7.6K views"}},"simpleText":"7.6K views"},"menu":{"menuRenderer":{"items":[{"menuServiceItemRenderer":{"text":{"runs":[{"text":"Add to queue"}]},"icon":{"iconType":"ADD_TO_QUEUE_TAIL"},"serviceEndpoint":{"clickTrackingParams":"CMEBEP6YBBgFIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA","commandMetadata":{"webCommandMetadata":{"sendPost":true}},"signalServiceEndpoint":{"signal":"CLIENT_SIGNAL","actions":[{"clickTrackingParams":"CMEBEP6YBBgFIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA","addToPlaylistCommand":{"openMiniplayer":false,"openListPanel":true,"videoId":"w7apAOKgBYI","listType":"PLAYLIST_EDIT_LIST_TYPE_QUEUE","onCreateListCommand":{"clickTrackingParams":"CMEBEP6YBBgFIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA","commandMetadata":{"webCommandMetadata":{"sendPost":true,"apiUrl":"/youtubei/v1/playlist/create"}},"createPlaylistServiceEndpoint":{"videoIds":["w7apAOKgBYI"],"params":"CAQ%3D"}},"videoIds":["w7apAOKgBYI"]}},{"clickTrackingParams":"CMEBEP6YBBgFIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA","openPopupAction":{"popup":{"notificationActionRenderer":{"responseText":{"simpleText":"Added to queue"},"trackingParams":"CMIBELlqIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA"}},"popupType":"TOAST"}}]}},"trackingParams":"CMEBEP6YBBgFIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA"}},{"menuServiceItemRenderer":{"text":{"runs":[{"text":"Share"}]},"icon":{"iconType":"SHARE"},"serviceEndpoint":{"clickTrackingParams":"CL0BEKQwGAIiEwjt1uq03oSMAxWZRnoFHdkuMMA=","commandMetadata":{"webCommandMetadata":{"sendPost":true,"apiUrl":"/youtubei/v1/share/get_share_panel"}},"shareEntityServiceEndpoint":{"serializedShareEntity":"Cgt3N2FwQU9LZ0JZSQ%3D%3D","commands":[{"clickTrackingParams":"CL0BEKQwGAIiEwjt1uq03oSMAxWZRnoFHdkuMMA=","openPopupAction":{"popup":{"unifiedSharePanelRenderer":{"trackingParams":"CMABEI5iIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA","showLoadingSpinner":true}},"popupType":"DIALOG","beReused":true}}]}},"trackingParams":"CL0BEKQwGAIiEwjt1uq03oSMAxWZRnoFHdkuMMA=","hasSeparator":true}}],"trackingParams":"CL0BEKQwGAIiEwjt1uq03oSMAxWZRnoFHdkuMMA=","accessibility":{"accessibilityData":{"label":"Action menu"}}}},"thumbnailOverlays":[{"thumbnailOverlayTimeStatusRenderer":{"text":{"accessibility":{"accessibilityData":{"label":"21 minutes, 40 seconds"}},"simpleText":"21:40"},"style":"DEFAULT"}},{"thumbnailOverlayToggleButtonRenderer":{"isToggled":false,"untoggledIcon":{"iconType":"WATCH_LATER"},"toggledIcon":{"iconType":"CHECK"},"untoggledTooltip":"Watch later","toggledTooltip":"Added","untoggledServiceEndpoint":{"clickTrackingParams":"CL8BEPnnAxgBIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA","commandMetadata":{"webCommandMetadata":{"sendPost":true,"apiUrl":"/youtubei/v1/browse/edit_playlist"}},"playlistEditEndpoint":{"playlistId":"WL","actions":[{"addedVideoId":"w7apAOKgBYI","action":"ACTION_ADD_VIDEO"}]}},"toggledServiceEndpoint":{"clickTrackingParams":"CL8BEPnnAxgBIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA","commandMetadata":{"webCommandMetadata":{"sendPost":true,"apiUrl":"/youtubei/v1/browse/edit_playlist"}},"playlistEditEndpoint":{"playlistId":"WL","actions":[{"action":"ACTION_REMOVE_VIDEO_BY_VIDEO_ID","removedVideoId":"w7apAOKgBYI"}]}},"untoggledAccessibility":{"accessibilityData":{"label":"Watch later"}},"toggledAccessibility":{"accessibilityData":{"label":"Added"}},"trackingParams":"CL8BEPnnAxgBIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA"}},{"thumbnailOverlayToggleButtonRenderer":{"untoggledIcon":{"iconType":"ADD_TO_QUEUE_TAIL"},"toggledIcon":{"iconType":"PLAYLIST_ADD_CHECK"},"untoggledTooltip":"Add to queue","toggledTooltip":"Added","untoggledServiceEndpoint":{"clickTrackingParams":"CL4BEMfsBBgCIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA","commandMetadata":{"webCommandMetadata":{"sendPost":true}},"signalServiceEndpoint":{"signal":"CLIENT_SIGNAL","actions":[{"clickTrackingParams":"CL4BEMfsBBgCIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA","addToPlaylistCommand":{"openMiniplayer":false,"openListPanel":true,"videoId":"w7apAOKgBYI","listType":"PLAYLIST_EDIT_LIST_TYPE_QUEUE","onCreateListCommand":{"clickTrackingParams":"CL4BEMfsBBgCIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA","commandMetadata":{"webCommandMetadata":{"sendPost":true,"apiUrl":"/youtubei/v1/playlist/create"}},"createPlaylistServiceEndpoint":{"videoIds":["w7apAOKgBYI"],"params":"CAQ%3D"}},"videoIds":["w7apAOKgBYI"]}}]}},"untoggledAccessibility":{"accessibilityData":{"label":"Add to queue"}},"toggledAccessibility":{"accessibilityData":{"label":"Added"}},"trackingParams":"CL4BEMfsBBgCIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA"}},{"thumbnailOverlayNowPlayingRenderer":{"text":{"runs":[{"text":"Now playing"}]}}}],"accessibility":{"accessibilityData":{"label":"Lecture 1(B): Sets and n-tuples - 21 minutes - Go to channel - Arizona Math Camp - 7.6K views - 6 years ago - play video"}}}},{"compactVideoRenderer":{"videoId":"Rqv3cXt8ZNU","thumbnail":{"thumbnails":[{"url":"https://i.ytimg.com/vi/Rqv3cXt8ZNU/hqdefault.jpg?sqp=-oaymwEiCKgBEF5IWvKriqkDFQgBFQAAAAAYASUAAMhCPQCAokN4AQ== rs=AOn4CLBi8Ddsdc25xsxckwVOF4II92Ab1Q","width":168,"height":94},{"url":"https://i.ytimg.com/vi/Rqv3cXt8ZNU/hqdefault.jpg?sqp=-oaymwEjCNACELwBSFryq4qpAxUIARUAAAAAGAElAADIQj0AgKJDeAE= rs=AOn4CLAxAVcJcX-MlGBP359gPu-NOr1K-g","width":336,"height":188}]},"title":{"accessibility":{"accessibilityData":{"label":"The unreasonable effectiveness of linear algebra. by Michael Penn 184,142 views 1 year ago 18 minutes"}},"simpleText":"The unreasonable effectiveness of linear algebra."},"longBylineText":{"runs":[{"text":"Michael Penn","navigationEndpoint":{"clickTrackingParams":"CLcBEKQwGAMiEwjt1uq03oSMAxWZRnoFHdkuMMAyB3JlbGF0ZWQ=","commandMetadata":{"webCommandMetadata":{"url":"/@MichaelPennMath","webPageType":"WEB_PAGE_TYPE_CHANNEL","rootVe":3611,"apiUrl":"/youtubei/v1/browse"}},"browseEndpoint":{"browseId":"UC6jM0RFkr4eSkzT5Gx0HOAw","canonicalBaseUrl":"/@MichaelPennMath"}}}]},"publishedTimeText":{"simpleText":"1 year ago"},"viewCountText":{"simpleText":"184,142 views"},"lengthText":{"accessibility":{"accessibilityData":{"label":"18 minutes, 4 seconds"}},"simpleText":"18:04"},"navigationEndpoint":{"clickTrackingParams":"CLcBEKQwGAMiEwjt1uq03oSMAxWZRnoFHdkuMMAyB3JlbGF0ZWRI9IjY2f6syY5SmgEFCAEQ-B0=","commandMetadata":{"webCommandMetadata":{"url":"/watch?v=Rqv3cXt8ZNU","webPageType":"WEB_PAGE_TYPE_WATCH","rootVe":3832}},"watchEndpoint":{"videoId":"Rqv3cXt8ZNU","nofollow":true,"watchEndpointSupportedOnesieConfig":{"html5PlaybackOnesieConfig":{"commonConfig":{"url":"https://rr1---sn-uxaxiv0nxx5q-nv4l.googlevideo.com/initplayback?source=youtube oeis=1 c=WEB oad=3200 ovd=3200 oaad=11000 oavd=11000 ocs=700 oewis=1 oputc=1 ofpcc=1 msp=1 odepv=1 onvi=1 id=46abf7717b7c64d5 ip=2a02%3A27aa%3A0%3A0%3A0%3A0%3A0%3Ad49 initcwndbps=1396250 mt=1741788795 oweuc="}}}}},"shortBylineText":{"runs":[{"text":"Michael Penn","navigationEndpoint":{"clickTrackingParams":"CLcBEKQwGAMiEwjt1uq03oSMAxWZRnoFHdkuMMAyB3JlbGF0ZWQ=","commandMetadata":{"webCommandMetadata":{"url":"/@MichaelPennMath","webPageType":"WEB_PAGE_TYPE_CHANNEL","rootVe":3611,"apiUrl":"/youtubei/v1/browse"}},"browseEndpoint":{"browseId":"UC6jM0RFkr4eSkzT5Gx0HOAw","canonicalBaseUrl":"/@MichaelPennMath"}}}]},"channelThumbnail":{"thumbnails":[{"url":"https://yt3.ggpht.com/9FLFQU9BbmNAHFZBGHrbSH12dJWlKhfaGltsfS5JgrVgugMMGiN78jJ-39iifZNK_3pF0UofXg=s68-c-k-c0x00ffffff-no-rj","width":68,"height":68}]},"ownerBadges":[{"metadataBadgeRenderer":{"icon":{"iconType":"CHECK_CIRCLE_THICK"},"style":"BADGE_STYLE_TYPE_VERIFIED","tooltip":"Verified","trackingParams":"CLcBEKQwGAMiEwjt1uq03oSMAxWZRnoFHdkuMMA=","accessibilityData":{"label":"Verified"}}}],"trackingParams":"CLcBEKQwGAMiEwjt1uq03oSMAxWZRnoFHdkuMMBA1cnx25fu_dVG","shortViewCountText":{"accessibility":{"accessibilityData":{"label":"184K views"}},"simpleText":"184K views"},"menu":{"menuRenderer":{"items":[{"menuServiceItemRenderer":{"text":{"runs":[{"text":"Add to queue"}]},"icon":{"iconType":"ADD_TO_QUEUE_TAIL"},"serviceEndpoint":{"clickTrackingParams":"CLsBEP6YBBgGIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA","commandMetadata":{"webCommandMetadata":{"sendPost":true}},"signalServiceEndpoint":{"signal":"CLIENT_SIGNAL","actions":[{"clickTrackingParams":"CLsBEP6YBBgGIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA","addToPlaylistCommand":{"openMiniplayer":false,"openListPanel":true,"videoId":"Rqv3cXt8ZNU","listType":"PLAYLIST_EDIT_LIST_TYPE_QUEUE","onCreateListCommand":{"clickTrackingParams":"CLsBEP6YBBgGIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA","commandMetadata":{"webCommandMetadata":{"sendPost":true,"apiUrl":"/youtubei/v1/playlist/create"}},"createPlaylistServiceEndpoint":{"videoIds":["Rqv3cXt8ZNU"],"params":"CAQ%3D"}},"videoIds":["Rqv3cXt8ZNU"]}},{"clickTrackingParams":"CLsBEP6YBBgGIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA","openPopupAction":{"popup":{"notificationActionRenderer":{"responseText":{"simpleText":"Added to queue"},"trackingParams":"CLwBELlqIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA"}},"popupType":"TOAST"}}]}},"trackingParams":"CLsBEP6YBBgGIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA"}},{"menuServiceItemRenderer":{"text":{"runs":[{"text":"Share"}]},"icon":{"iconType":"SHARE"},"serviceEndpoint":{"clickTrackingParams":"CLcBEKQwGAMiEwjt1uq03oSMAxWZRnoFHdkuMMA=","commandMetadata":{"webCommandMetadata":{"sendPost":true,"apiUrl":"/youtubei/v1/share/get_share_panel"}},"shareEntityServiceEndpoint":{"serializedShareEntity":"CgtScXYzY1h0OFpOVQ%3D%3D","commands":[{"clickTrackingParams":"CLcBEKQwGAMiEwjt1uq03oSMAxWZRnoFHdkuMMA=","openPopupAction":{"popup":{"unifiedSharePanelRenderer":{"trackingParams":"CLoBEI5iIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA","showLoadingSpinner":true}},"popupType":"DIALOG","beReused":true}}]}},"trackingParams":"CLcBEKQwGAMiEwjt1uq03oSMAxWZRnoFHdkuMMA=","hasSeparator":true}}],"trackingParams":"CLcBEKQwGAMiEwjt1uq03oSMAxWZRnoFHdkuMMA=","accessibility":{"accessibilityData":{"label":"Action menu"}}}},"thumbnailOverlays":[{"thumbnailOverlayTimeStatusRenderer":{"text":{"accessibility":{"accessibilityData":{"label":"18 minutes, 4 seconds"}},"simpleText":"18:04"},"style":"DEFAULT"}},{"thumbnailOverlayToggleButtonRenderer":{"isToggled":false,"untoggledIcon":{"iconType":"WATCH_LATER"},"toggledIcon":{"iconType":"CHECK"},"untoggledTooltip":"Watch later","toggledTooltip":"Added","untoggledServiceEndpoint":{"clickTrackingParams":"CLkBEPnnAxgCIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA","commandMetadata":{"webCommandMetadata":{"sendPost":true,"apiUrl":"/youtubei/v1/browse/edit_playlist"}},"playlistEditEndpoint":{"playlistId":"WL","actions":[{"addedVideoId":"Rqv3cXt8ZNU","action":"ACTION_ADD_VIDEO"}]}},"toggledServiceEndpoint":{"clickTrackingParams":"CLkBEPnnAxgCIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA","commandMetadata":{"webCommandMetadata":{"sendPost":true,"apiUrl":"/youtubei/v1/browse/edit_playlist"}},"playlistEditEndpoint":{"playlistId":"WL","actions":[{"action":"ACTION_REMOVE_VIDEO_BY_VIDEO_ID","removedVideoId":"Rqv3cXt8ZNU"}]}},"untoggledAccessibility":{"accessibilityData":{"label":"Watch later"}},"toggledAccessibility":{"accessibilityData":{"label":"Added"}},"trackingParams":"CLkBEPnnAxgCIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA"}},{"thumbnailOverlayToggleButtonRenderer":{"untoggledIcon":{"iconType":"ADD_TO_QUEUE_TAIL"},"toggledIcon":{"iconType":"PLAYLIST_ADD_CHECK"},"untoggledTooltip":"Add to queue","toggledTooltip":"Added","untoggledServiceEndpoint":{"clickTrackingParams":"CLgBEMfsBBgDIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA","commandMetadata":{"webCommandMetadata":{"sendPost":true}},"signalServiceEndpoint":{"signal":"CLIENT_SIGNAL","actions":[{"clickTrackingParams":"CLgBEMfsBBgDIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA","addToPlaylistCommand":{"openMiniplayer":false,"openListPanel":true,"videoId":"Rqv3cXt8ZNU","listType":"PLAYLIST_EDIT_LIST_TYPE_QUEUE","onCreateListCommand":{"clickTrackingParams":"CLgBEMfsBBgDIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA","commandMetadata":{"webCommandMetadata":{"sendPost":true,"apiUrl":"/youtubei/v1/playlist/create"}},"createPlaylistServiceEndpoint":{"videoIds":["Rqv3cXt8ZNU"],"params":"CAQ%3D"}},"videoIds":["Rqv3cXt8ZNU"]}}]}},"untoggledAccessibility":{"accessibilityData":{"label":"Add to queue"}},"toggledAccessibility":{"accessibilityData":{"label":"Added"}},"trackingParams":"CLgBEMfsBBgDIhMI7dbqtN6EjAMVmUZ6BR3ZLjDA"}},{"thumbnailOverlayNowPlayingRenderer":{"text":{"runs":[{"text":"Now playing"}]}}}],"accessibility":{"accessibilityData":{"label":"The unreasonable effectiveness of linear algebra. - 18 minutes - Go to channel - Michael Penn - 184K views - 1 year ago - play video"}}}},{"lockupViewModel":{"contentImage":{"collectionThumbnailViewModel":{"primaryThumbnail":{"thumbnailViewModel":{"image":{"sources":[{"url":"https://i.ytimg.com/vi/Tyt9SlcZSk0/hqdefault.jpg?sqp=-oaymwEWCKgBEF5IWvKriqkDCQgBFQAAiEIYAQ== rs=AOn4CLD29mQ6jEd6UCsB8OeWidkT8Uh1eQ","width":168,"height":94},{"url":"https://i.ytimg.com/vi/Tyt9SlcZSk0/hqdefault.jpg?sqp=-oaymwEXCNACELwBSFryq4qpAwkIARUAAIhCGAE= rs=AOn4CLDqCNhuTEqwUyrmzOhH4l_qmTEsCQ","width":336,"height":188}]},"overlays":[{"thumbnailOverlayBadgeViewModel":{"thumbnailBadges":[{"thumbnailBadgeViewModel":{"icon":{"sources":[{"clientResource":{"imageName":"PLAYLISTS"}}]},"text":"42 videos","badgeStyle":"THUMBNAIL_OVERLAY_BADGE_STYLE_DEFAULT","backgroundColor":{"lightTheme":2631459,"darkTheme":2631459}}}],"position":"THUMBNAIL_OVERLAY_BADGE_POSITION_BOTTOM_END"}},{"thumbnailHoverOverlayViewModel":{"icon":{"sources":[{"clientResource":{"imageName":"PLAY_ALL"}}]},"text":{"content":"Play all","styleRuns":[{"startIndex":0,"length":8}]},"style":"THUMBNAIL_HOVER_OVERLAY_STYLE_COVER"}}],"backgroundColor":{"lightTheme":4144439,"darkTheme":4144439}}},"stackColor":{"lightTheme":12564903,"darkTheme":9209722}}},"metadata":{"lockupMetadataViewModel":{"title":{"content":"Linear Algebra"},"metadata":{"contentMetadataViewModel":{"metadataRows":[{"metadataParts":[{"text":{"content":"Math for Thought

#############################












Content Report
Youtor.org / YTube video Downloader © 2025

created by www.youtor.org