<div class="xblock xblock-public_view xblock-public_view-vertical" data-init="VerticalStudentView" data-course-id="course-v1:OCW+18.06SC+2T2019" data-request-token="f0206040ed2f11eeb8fa16ffd59041e1" data-graded="False" data-usage-id="block-v1:OCW+18.06SC+2T2019+type@vertical+block@236c988e7fcd407283b94461363330bb" data-block-type="vertical" data-runtime-class="LmsRuntime" data-runtime-version="1" data-has-score="False">
<h2 class="hd hd-2 unit-title">1.9 Session Overview</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:OCW+18.06SC+2T2019+type@html+block@71cda85181f640eaa4a0cf730a6b9b1c">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-init="XBlockToXModuleShim" data-course-id="course-v1:OCW+18.06SC+2T2019" data-request-token="f0206040ed2f11eeb8fa16ffd59041e1" data-graded="False" data-usage-id="block-v1:OCW+18.06SC+2T2019+type@html+block@71cda85181f640eaa4a0cf730a6b9b1c" data-block-type="html" data-runtime-class="LmsRuntime" data-runtime-version="1" data-has-score="False">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<div style="float: left; margin: 6px 20px auto 0px;"><img height="100" width="100" src="/assets/courseware/v1/31cca1ea33c9f05675e14203c13c7166/asset-v1:OCW+18.06SC+2T2019+type@asset+block/18.06_1.9_Session_Overview_10.13.43_AM.jpg" alt="Figure excerpted from 'Introduction to Linear Algebra' by G.S. Strang" /></div>
<p>We describe all solutions to <em>Ax</em> = b based on the free variables and special solutions encoded in the reduced form <em>R.</em></p>
<p></p>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-init="VerticalStudentView" data-course-id="course-v1:OCW+18.06SC+2T2019" data-request-token="f0206040ed2f11eeb8fa16ffd59041e1" data-graded="False" data-usage-id="block-v1:OCW+18.06SC+2T2019+type@vertical+block@7206fbbafc7f451995a51c6a22862180" data-block-type="vertical" data-runtime-class="LmsRuntime" data-runtime-version="1" data-has-score="False">
<h2 class="hd hd-2 unit-title">Lecture Video</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:OCW+18.06SC+2T2019+type@video+block@f2d7fbf28a7f4c03aa9016abc7716a94">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-init="XBlockToXModuleShim" data-course-id="course-v1:OCW+18.06SC+2T2019" data-request-token="f0206040ed2f11eeb8fa16ffd59041e1" data-graded="False" data-usage-id="block-v1:OCW+18.06SC+2T2019+type@video+block@f2d7fbf28a7f4c03aa9016abc7716a94" data-block-type="video" data-runtime-class="LmsRuntime" data-runtime-version="1" data-has-score="False">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Solving Ax = b: Row Reduced Form R</h3>
<div
id="video_f2d7fbf28a7f4c03aa9016abc7716a94"
class="video closed"
data-metadata='{"completionPercentage": 0.95, "ytMetadataEndpoint": "", "prioritizeHls": false, "transcriptAvailableTranslationsUrl": "/courses/course-v1:OCW+18.06SC+2T2019/xblock/block-v1:OCW+18.06SC+2T2019+type@video+block@f2d7fbf28a7f4c03aa9016abc7716a94/handler/transcript/available_translations", "generalSpeed": 1.0, "saveStateUrl": "/courses/course-v1:OCW+18.06SC+2T2019/xblock/block-v1:OCW+18.06SC+2T2019+type@video+block@f2d7fbf28a7f4c03aa9016abc7716a94/handler/xmodule_handler/save_user_state", "savedVideoPosition": 0.0, "ytApiUrl": "https://www.youtube.com/iframe_api", "end": 0.0, "poster": null, "ytTestTimeout": 1500, "completionEnabled": false, "streams": "1.00:9Q1q7s1jTzU", "publishCompletionUrl": "/courses/course-v1:OCW+18.06SC+2T2019/xblock/block-v1:OCW+18.06SC+2T2019+type@video+block@f2d7fbf28a7f4c03aa9016abc7716a94/handler/publish_completion", "showCaptions": "true", "start": 0.0, "sources": ["https://archive.org/download/MIT18.06S05_MP4/08.mp4"], "autoplay": false, "captionDataDir": null, "recordedYoutubeIsAvailable": true, "saveStateEnabled": false, "transcriptLanguages": {"en": "English"}, "transcriptTranslationUrl": "/courses/course-v1:OCW+18.06SC+2T2019/xblock/block-v1:OCW+18.06SC+2T2019+type@video+block@f2d7fbf28a7f4c03aa9016abc7716a94/handler/transcript/translation/__lang__", "autoAdvance": false, "autohideHtml5": false, "lmsRootURL": "https://openlearninglibrary.mit.edu", "speed": null, "transcriptLanguage": "en", "duration": 0.0}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="f2d7fbf28a7f4c03aa9016abc7716a94"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_f2d7fbf28a7f4c03aa9016abc7716a94">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_f2d7fbf28a7f4c03aa9016abc7716a94">
<div class="wrapper-download-video">
<h4 class="hd hd-5">Video</h4>
<a class="btn-link video-sources video-download-button" href="https://archive.org/download/MIT18.06S05_MP4/08.mp4">
Download video file
</a>
</div>
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:OCW+18.06SC+2T2019/xblock/block-v1:OCW+18.06SC+2T2019+type@video+block@f2d7fbf28a7f4c03aa9016abc7716a94/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:OCW+18.06SC+2T2019/xblock/block-v1:OCW+18.06SC+2T2019+type@video+block@f2d7fbf28a7f4c03aa9016abc7716a94/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:OCW+18.06SC+2T2019+type@html+block@5bec6fd75d3a4e9e87254c4809d5de2a">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-init="XBlockToXModuleShim" data-course-id="course-v1:OCW+18.06SC+2T2019" data-request-token="f0206040ed2f11eeb8fa16ffd59041e1" data-graded="False" data-usage-id="block-v1:OCW+18.06SC+2T2019+type@html+block@5bec6fd75d3a4e9e87254c4809d5de2a" data-block-type="html" data-runtime-class="LmsRuntime" data-runtime-version="1" data-has-score="False">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><a href="/assets/courseware/v1/eb58941e5a08c92caa883090a6ac3eb7/asset-v1:OCW+18.06SC+2T2019+type@asset+block/18.06_1.9_LecTrans.pdf" target="[object Object]">Lecture video transcript (PDF)</a></p>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-init="VerticalStudentView" data-course-id="course-v1:OCW+18.06SC+2T2019" data-request-token="f0206040ed2f11eeb8fa16ffd59041e1" data-graded="False" data-usage-id="block-v1:OCW+18.06SC+2T2019+type@vertical+block@d72c0074ea97489c8dbb786413aca3fd" data-block-type="vertical" data-runtime-class="LmsRuntime" data-runtime-version="1" data-has-score="False">
<h2 class="hd hd-2 unit-title">Lecture Summary and Reading</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:OCW+18.06SC+2T2019+type@html+block@4759c1fcc2914b008cd015676dedf263">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-init="XBlockToXModuleShim" data-course-id="course-v1:OCW+18.06SC+2T2019" data-request-token="f0206040ed2f11eeb8fa16ffd59041e1" data-graded="False" data-usage-id="block-v1:OCW+18.06SC+2T2019+type@html+block@4759c1fcc2914b008cd015676dedf263" data-block-type="html" data-runtime-class="LmsRuntime" data-runtime-version="1" data-has-score="False">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<h3>Lecture Summary</h3>
<ul>
<li><a href="/assets/courseware/v1/f84a01e06b6fd0c34cd686c6b143ef2e/asset-v1:OCW+18.06SC+2T2019+type@asset+block/18.06_1.9_LecSum.pdf" target="[object Object]">Solving <em>A</em>x = b: Row Reduced Form R (PDF)</a></li>
</ul>
<h3>Suggested Reading</h3>
<ul>
<li>Read Section 3.3 through 3.4 in the 4<sup>th</sup> or 5<sup>th</sup> edition.</li>
</ul>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-init="VerticalStudentView" data-course-id="course-v1:OCW+18.06SC+2T2019" data-request-token="f0206040ed2f11eeb8fa16ffd59041e1" data-graded="False" data-usage-id="block-v1:OCW+18.06SC+2T2019+type@vertical+block@86a59c13a8ab425c8bc065ff6dd1a703" data-block-type="vertical" data-runtime-class="LmsRuntime" data-runtime-version="1" data-has-score="False">
<h2 class="hd hd-2 unit-title">Recitation Video</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:OCW+18.06SC+2T2019+type@video+block@c4b14ac63d414e5496dd56e95c866e89">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-init="XBlockToXModuleShim" data-course-id="course-v1:OCW+18.06SC+2T2019" data-request-token="f0206040ed2f11eeb8fa16ffd59041e1" data-graded="False" data-usage-id="block-v1:OCW+18.06SC+2T2019+type@video+block@c4b14ac63d414e5496dd56e95c866e89" data-block-type="video" data-runtime-class="LmsRuntime" data-runtime-version="1" data-has-score="False">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Solving Ax = b</h3>
<div
id="video_c4b14ac63d414e5496dd56e95c866e89"
class="video closed"
data-metadata='{"completionPercentage": 0.95, "ytMetadataEndpoint": "", "prioritizeHls": false, "transcriptAvailableTranslationsUrl": "/courses/course-v1:OCW+18.06SC+2T2019/xblock/block-v1:OCW+18.06SC+2T2019+type@video+block@c4b14ac63d414e5496dd56e95c866e89/handler/transcript/available_translations", "generalSpeed": 1.0, "saveStateUrl": "/courses/course-v1:OCW+18.06SC+2T2019/xblock/block-v1:OCW+18.06SC+2T2019+type@video+block@c4b14ac63d414e5496dd56e95c866e89/handler/xmodule_handler/save_user_state", "savedVideoPosition": 0.0, "ytApiUrl": "https://www.youtube.com/iframe_api", "end": 0.0, "poster": null, "ytTestTimeout": 1500, "completionEnabled": false, "streams": "1.00:75Q0ZN2njGQ", "publishCompletionUrl": "/courses/course-v1:OCW+18.06SC+2T2019/xblock/block-v1:OCW+18.06SC+2T2019+type@video+block@c4b14ac63d414e5496dd56e95c866e89/handler/publish_completion", "showCaptions": "true", "start": 0.0, "sources": ["https://archive.org/download/MIT18.06SCF11/MIT18_06SC_110714_M3_300k.mp4"], "autoplay": false, "captionDataDir": null, "recordedYoutubeIsAvailable": true, "saveStateEnabled": false, "transcriptLanguages": {"en": "English"}, "transcriptTranslationUrl": "/courses/course-v1:OCW+18.06SC+2T2019/xblock/block-v1:OCW+18.06SC+2T2019+type@video+block@c4b14ac63d414e5496dd56e95c866e89/handler/transcript/translation/__lang__", "autoAdvance": false, "autohideHtml5": false, "lmsRootURL": "https://openlearninglibrary.mit.edu", "speed": null, "transcriptLanguage": "en", "duration": 0.0}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="c4b14ac63d414e5496dd56e95c866e89"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_c4b14ac63d414e5496dd56e95c866e89">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_c4b14ac63d414e5496dd56e95c866e89">
<div class="wrapper-download-video">
<h4 class="hd hd-5">Video</h4>
<a class="btn-link video-sources video-download-button" href="https://archive.org/download/MIT18.06SCF11/MIT18_06SC_110714_M3_300k.mp4">
Download video file
</a>
</div>
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:OCW+18.06SC+2T2019/xblock/block-v1:OCW+18.06SC+2T2019+type@video+block@c4b14ac63d414e5496dd56e95c866e89/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:OCW+18.06SC+2T2019/xblock/block-v1:OCW+18.06SC+2T2019+type@video+block@c4b14ac63d414e5496dd56e95c866e89/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:OCW+18.06SC+2T2019+type@html+block@254f021f77d84fec82fbdd9e3258a592">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-init="XBlockToXModuleShim" data-course-id="course-v1:OCW+18.06SC+2T2019" data-request-token="f0206040ed2f11eeb8fa16ffd59041e1" data-graded="False" data-usage-id="block-v1:OCW+18.06SC+2T2019+type@html+block@254f021f77d84fec82fbdd9e3258a592" data-block-type="html" data-runtime-class="LmsRuntime" data-runtime-version="1" data-has-score="False">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><a href="/assets/courseware/v1/220fd26c52d29e2f3f038678217aaa70/asset-v1:OCW+18.06SC+2T2019+type@asset+block/18.06_1.9_RecitationTrans.pdf" target="[object Object]">Recitation video transcript (PDF)</a></p>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-init="VerticalStudentView" data-course-id="course-v1:OCW+18.06SC+2T2019" data-request-token="f0206040ed2f11eeb8fa16ffd59041e1" data-graded="False" data-usage-id="block-v1:OCW+18.06SC+2T2019+type@vertical+block@7f46af9c1abe40d98e0a338027d80da8" data-block-type="vertical" data-runtime-class="LmsRuntime" data-runtime-version="1" data-has-score="False">
<h2 class="hd hd-2 unit-title">Problems and Solutions</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:OCW+18.06SC+2T2019+type@html+block@368b31a24b9b4151acdfe32aa2c5fef4">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-init="XBlockToXModuleShim" data-course-id="course-v1:OCW+18.06SC+2T2019" data-request-token="f0206040ed2f11eeb8fa16ffd59041e1" data-graded="False" data-usage-id="block-v1:OCW+18.06SC+2T2019+type@html+block@368b31a24b9b4151acdfe32aa2c5fef4" data-block-type="html" data-runtime-class="LmsRuntime" data-runtime-version="1" data-has-score="False">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p>Work the problems on your own and check your answers when you're done.</p>
<p><a href="/assets/courseware/v1/d885df8a1aeeb1f514f9d89f5618f85e/asset-v1:OCW+18.06SC+2T2019+type@asset+block/18.06_1.9_Exercises.pdf" target="[object Object]">Exercises on solving <em>A</em>x = b and row reduced form <em>R</em> (PDF)</a></p>
<p><a href="/assets/courseware/v1/65d512eb10210845ae9e71b06fa4e293/asset-v1:OCW+18.06SC+2T2019+type@asset+block/18.06_1.9_Solutions.pdf" target="[object Object]">Solutions to solving <em>A</em>x = b and row reduced form <em>R</em> (PDF)</a></p>
</div>
</div>
</div>
</div>