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<h2 class="hd hd-2 unit-title">Overview and Prerequisites</h2>
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<h3><img height="275" width="320" src="/assets/courseware/v1/ad495cddb0b1cc7d6c6de7b43c9ad0ed/asset-v1:OCW+18.06SC+2T2019+type@asset+block/18-06scf11.jpg" alt="Multi-colored cubes arranged to create the illusion of hexagons" /></h3>
<p class="caption"><span style="color: #999999;"> This optical illusion was seen on a restaurant in Paris, and <a href="https://ocw.mit.edu/courses/mathematics/18-06sc-linear-algebra-fall-2011/HexagonArt.m" target="[object Object]">coded in MATLAB</a> by <a href="http://news.mit.edu/2012/mit-opencourseware-publishes-linear-algebra-in-innovative-ocw-scholar-format" target="[object Object]">Shev Macnamara.</a> Each color can become the tops of the cubes if you look at them correctly.</span></p>
<h3>Course Overview</h3>
<p>This course covers matrix theory and linear algebra, emphasizing topics useful in other disciplines. Linear algebra is a branch of mathematics that studies systems of linear equations and the properties of matrices. The concepts of linear algebra are extremely useful in physics, economics and social sciences, natural sciences, and engineering. Due to its broad range of applications, linear algebra is one of the most widely taught subjects in college-level mathematics (and increasingly in high school).</p>
<h3>Prerequisites</h3>
<p><span style="text-decoration: underline;"><em><a href="https://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/" target="[object Object]">18.02 Multivariable Calculus (OCW)</a></em></span> is a formal prerequisite for MIT students wishing to enroll in 18.06 Linear Algebra, but knowledge of calculus is not required to learn the subject.</p>
<p>To succeed in this course you will need to be comfortable with vectors, matrices, and three-dimensional coordinate systems. This material is presented in the first few lectures of 18.02 Multivariable Calculus, and again here.</p>
<p>The basic operations of linear algebra are those you learned in grade school – addition and multiplication to produce "linear combinations." But with vectors, we move into four-dimensional space and n-dimensional space!</p>
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<h2 class="hd hd-2 unit-title">Goals</h2>
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<p>After successfully completing the course, you will have a good understanding of the following topics and their applications:</p>
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<li>Systems of linear equations</li>
<li>Row reduction and echelon forms</li>
<li>Matrix operations, including inverses</li>
<li>Block matrices</li>
<li>Linear dependence and independence</li>
<li>Subspaces and bases and dimensions</li>
<li>Orthogonal bases and orthogonal projections</li>
<li>Gram-Schmidt process</li>
<li>Linear models and least-squares problems</li>
<li>Determinants and their properties</li>
<li>Cramer's Rule</li>
<li>Eigenvalues and eigenvectors</li>
<li>Diagonalization of a matrix</li>
<li>Symmetric matrices</li>
<li>Positive definite matrices</li>
<li>Similar matrices</li>
<li>Linear transformations</li>
<li>Singular Value Decomposition</li>
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<h2 class="hd hd-2 unit-title">Format</h2>
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<p>This course, designed for independent study, has been organized to follow the sequence of topics covered in an MIT course on Linear Algebra. The content is organized into three major units:</p>
<p>UNIT I. <em>A</em>x = b and the Four Subspaces</p>
<p>UNIT II. Least Squares, Determinants and Eigenvalues</p>
<p>UNIT III. Positive Definite Matrices and Applications</p>
<p>Each unit has been further divided into a sequence of sessions that cover an amount you might expect to complete in one sitting. Each session has a video lecture on the topic, accompanied by a lecture summary. For further study, there are suggested readings in Professor Strang's textbook (both the 4th and 5th editions):</p>
<p><a href="http://www.amazon.com/exec/obidos/ASIN/0980232716/ref=nosim/mitopencourse-20"><img alt="Buy at Amazon" src="https://ocw.mit.edu/images/a_logo_17.gif" border="0" align="absMiddle" /></a> Strang, Gilbert. Introduction to Linear Algebra. 4th ed. Wellesley, MA: <a href="http://www.wellesleycambridge.com/">Wellesley-Cambridge Press</a>, February 2009. ISBN: 9780980232714.</p>
<p><a href="http://www.amazon.com/exec/obidos/ASIN/0980232775/ref=nosim/mitopencourse-20"><img alt="Buy at Amazon" src="https://ocwcms.mit.edu/images/a_logo_17.gif" border="0" align="absmiddle" /></a><em> </em>Strang, Gilbert. Introduction to Linear Algebra. 5th ed. Wellesley, MA: <a href="http://www.wellesleycambridge.com/">Wellesley-Cambridge Press</a>, February 2016. ISBN: 9780980232776.</p>
<p>To help guide your learning, you will see how problem solving is taught by an experienced MIT Recitation instructor (six of the Problem Solving Videos are also available in Mandarin Chinese).</p>
<p>Finally, within each unit you will be presented with sets of problems at strategic points, so you can test your understanding of the material.</p>
<p style="text-align: left;">To access, click on the navigation links in the course module to display the sessions inside each of the three units. <br /> <br /> MIT expects its students to spend about 150 hours on this course. More than half of that time is spent preparing for class and doing assignments. It's difficult to estimate how long it will take you to complete the course, but you can probably expect to spend an hour or more working through each individual session.</p>
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<h2 class="hd hd-2 unit-title">Meet the Team</h2>
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<p><img width="147px" height="220px" alt="A woman in a black coat with a blue necklace stands in front of a pond." src="/assets/courseware/v1/6c67eb3701b705a171265b54354f4997/asset-v1:OCW+18.06SC+2T2019+type@asset+block/martina_photo.jpg" style="padding: 12px 0px 15px 15px; float: right;" /> <strong>Martina Balagovic</strong><br /> <em>Hi, I am Martina. By now, I taught linear algebra classes in three universities and in many different flavors. I love algebra, and this is probably my favorite class to teach. It gives students the chance to soar into rigorous, abstract algebraic thinking, while being firmly grounded by its many examples. This way one can reason abstractly and then check one's intuition and reasoning on small, very concrete examples. At the same time, one can explore numerous applications in engineering and computer science. I hope the examples we prepared for you will be useful!</em></p>
<p>Martina Balagovic grew up in Zagreb, Croatia. She got her undergraduate degree from the University of Zagreb. She was a graduate student at MIT from 2007, earning a Ph.D. degree in 2011. She is currently a postdoc at the University of York in the UK. She is fascinated by algebra and does research in representation theory.</p>
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<p><strong>Linan Chen</strong><br /> <em>Hello, my name is Linan and I'm glad to meet you here. When I was a graduate student at MIT, I had this great opportunity to make some video sessions on solving linear algebra problems for Course 18.06. So, here I am, with the exercises I choose for you. I would like to show you how simple pictures of lines intersecting can give rise to matrices, and how matrices can help with problems from other subjects such as solving differential equations or calculating volumes. Besides, I believe you also notice from these videos that drawing the related picture can often speed up the problem solving process, and it's important that you practice the skill of drawing such pictures. So, have fun with these exercises, and I hope you find them good supplements to the lectures.</em></p>
<p>Linan Chen comes from Shenyang, a city in the northeast of China. After completing her B.A. in Mathematics from Tsinghua University, she continued her graduate study at MIT where she obtained her Ph.D. in Mathematics in 2011. Linan has been an instructor for various math courses, and for her teaching effort, she was awarded the Charles and Holly Housman Award for Excellence in Teaching from the Department of Mathematics at MIT in 2011. Linan also created six recitation sessions for this course in Mandarin Chinese. You can find them in the following sessions:</p>
<img width="185px" height="240px" src="/assets/courseware/v1/5fec5a8ec8d5e8369e97890fe856cef5/asset-v1:OCW+18.06SC+2T2019+type@asset+block/linan_photo.jpg" alt="A woman with black hair in a black sweater stands in front of a blackboard. She has her hand on her hip." style="padding: 10px 30px 15px 0px; float: left;" />
<ul>
<li>The Geometry of Linear Algebra</li>
<li>Transposes, Permutations, Vector Spaces</li>
<li>Determinant Formulas and Cofactors</li>
<li>Cramer's Rule, Inverse Matrix and Volume</li>
<li>Differential Equations and exp(At)</li>
<li>Exam 2 Review</li>
</ul>
<p>Linan is currently a Postdoctoral Fellow at McGill University, and her main research interest lies in probability theory, as well as its applications in other disciplines.</p>
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<p><img src="/assets/courseware/v1/ea85719fa36f9f948fb51a71ede753df/asset-v1:OCW+18.06SC+2T2019+type@asset+block/Ben_photo.jpg" alt="A headshot of a young man with red hair and glasses." style="padding: 12px 0px 15px 15px; float: right; width: 163px; height: 201px;" /></p>
<p><strong>Benjamin Harris</strong> <br /> <em>Hey, I'm Ben and I taught recitations of 18.06: Linear Algebra at MIT in the spring of 2009 and the spring of 2010. In 18.06, I enjoyed helping students take apart matrices, whether it is with the PLU-factorization, the QR-decomposition, or eigenvalues and eigenvectors. Matrices and linear transformations occur naturally throughout science and math. I hope Gilbert Strang's lectures and our recitation videos help you to pull apart these mathematical objects.</em></p>
<p>Benjamin Harris is currently a postdoctoral researcher at Louisiana State University. He obtained his Ph.D. in Mathematics from MIT in 2011. His research concerns Lie groups and their representations. More specifically, he is interested in wave front cycles of tempered representations, Fourier transforms of nilpotent coadjoint orbits, irreducible characters, and branching laws for discrete series.</p>
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<p><img width="153" height="210" src="/assets/courseware/v1/d7c58e0e0f19e83dcaf3dd60d148a4ea/asset-v1:OCW+18.06SC+2T2019+type@asset+block/Ana_photo.jpg" alt="A young woman smiles brightly at the camera. She is sitting in front of some green plants and is wearing a bright red and white shirt." style="padding: 0px 15px 15px 0px; float: left;" /><strong>Ana Rita Pires</strong><br /> <em>My name is Ana. In 2011 when I started as a postdoc at Cornell University, my first teaching assignment was...Linear Algebra! I have taught 18.06 several times here at MIT, and Linear Algebra might be my favorite subject to teach, because it has so many neat ideas and interesting applications, all the while being fairly self-contained.</em></p>
<p>Ana grew up in Portugal and got her undergraduate degree there. She came to MIT in 2005 as a graduate student in mathematics and started her research on symplectic geometry.</p>
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<p><img width="207" height="190" src="/assets/courseware/v1/2d9c460e247c532d1ee81cfc992bdf90/asset-v1:OCW+18.06SC+2T2019+type@asset+block/David_Photo.jpg" alt="A young man with black hair is standing beside a chalkboard with equations on it. He is wearing a sweater with grey and yellow horizontal stripes. His hands are in his pockets." style="padding: 12px 0px 15px 15px; float: right;" /><strong>David Shirokoff</strong> <br /> <em>Hi, I'm Dave and welcome to 18.06. This course is a great backbone math class for any engineering or science student. For instance, many of the concepts covered will reappear in upper year courses and provide a stepping stone towards understanding "how things work". I hope you enjoy the problems in the videos, and that they help provide further insight into the material!</em></p>
<p>David Shirokoff grew up in Welland (ON) Canada and completed his undergrad at the University of Toronto. He recently finished his Ph.D. at MIT in applied math under the supervision of Ruben Rosales. His research interests are in applied differential equations, dynamical systems and numerical methods.</p>
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<p><img width="147px" height="220px;" src="/assets/courseware/v1/58f41e4fb3301c5132a2a04ceec246f1/asset-v1:OCW+18.06SC+2T2019+type@asset+block/Nikola_photo.jpg" alt="A young man with black hair and a striped shirt stands in the foreground." style="padding: 0px 15px 15px 0px; float: left;" /><strong>Nikola Kamburov</strong><br /> <em>Hi, I am Nikola. I hope you find these recitation videos, supplementing Prof. Strang's superb linear algebra lectures, helpful and illuminating. Good luck with your studies!</em></p>
<p>Nikola Kamburov is a Ph.D. student in the Department of Mathematics at MIT. He grew up in Bulgaria and he earned his bachelor's degree in mathematics from Princeton University. His research focuses on nonlinear partial differential equations.</p>
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<p><img width="207" height="200" src="/assets/courseware/v1/74e2192ff944a4ff4bee66914f8c2406/asset-v1:OCW+18.06SC+2T2019+type@asset+block/HeidiBurgiel.jpg" alt="Heidi Burgiel" style="padding: 12px 0px 15px 20px; float: right;" /><strong>Heidi Burgiel</strong> <br /> Heidi is an Instructional Designer at Lasell College and helps faculty adopt new tools and technologies to improve instruction. She earned her Ph.D. in Mathematics from the University of Washington, her bachelor's degree in mathematics from MIT and has recently earned a Master of Education from the Harvard Graduate School of Education. Heidi wrote all of the lecture summaries for this course. </p>
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