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Lecture 14: Orthogonal vectors and subspaces | Linear Algebra | Mathematics | MIT OpenCourseWare

Vectors are easier to understand when they’re described in terms of orthogonal bases. In addition, the Four Fundamental Subspaces are orthogonal to each other in pairs. If A is a rectangular matrix, Ax = b is often unsolvable. These video lectures of Pro

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Lecture 13: Quiz 1 review | Linear Algebra | Mathematics | MIT OpenCourseWare

By approaching what we’ve learned from new directions, the questions in this exam review session test the depth of your understanding. This unit reached the key ideas of subspaces—a higher level of linear algebra. These video lectures of Professor Gilb

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Lecture 12: Graphs, networks, incidence matrices | Linear Algebra | Mathematics | MIT OpenCourseWare

This session explores the linear algebra of electrical networks and the Internet, and sheds light on important results in graph theory. These video lectures of Professor Gilbert Strang teaching 18.06 were recorded in Fall 1999 and do not correspond precise

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Lecture 11: Matrix spaces; rank 1; small world graphs | Linear Algebra | Mathematics | MIT OpenCourseWare

As we learned last session, vectors don’t have to be lists of numbers. In this session we explore important new vector spaces while practicing the skills we learned in the old ones. Then we begin the application of matrices to the study of networks. Thes

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Lecture 10: The four fundamental subspaces | Linear Algebra | Mathematics | MIT OpenCourseWare

For some vectors b the equation Ax = b has solutions and for others it does not. Some vectors x are solutions to the equation Ax = 0 and some are not. To understand these equations we study the column space, nullspace, row space and left nullspace of the m

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Lecture 9: Independence, basis, and dimension | Linear Algebra | Mathematics | MIT OpenCourseWare

What does it mean for vectors to be independent? How does the idea of independence help us describe subspaces like the nullspace? A basis is a set of vectors, as few as possible, whose combinations produce all vectors in the space. The number of basis vect

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Lecture 8: Solving Ax = b: row reduced form R | Linear Algebra | Mathematics | MIT OpenCourseWare

When does Ax = b have solutions x, and how can we describe those solutions? We describe all solutions to Ax = b based on the free variables and special solutions encoded in the reduced form R. These video lectures of Professor Gilbert Strang teaching 18.06

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Lecture 7: Solving Ax = 0: pivot variables, special solutions | Linear Algebra | Mathematics | MIT OpenCourseWare

We apply the method of elimination to all matrices, invertible or not. Counting the pivots gives us the rank of the matrix. Further simplifying the matrix puts it in reduced row echelon form R and improves our description of the nullspace. These video lect

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