随笔分类 - 数学
摘要：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 matrix A.
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摘要：A basis is a set of vectors, as few as possible, whose combinations produce all vectors in the space. The number of basis vectors for a space equals the dimension of that space
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摘要：We describe all solutions to Ax = b based on the free variables and special solutions encoded in the reduced form R.
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摘要：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 null space.
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摘要：The column space of a matrix A tells us when the equation Ax = b will have a solution x. The null space of A tells us which values of x solve the equation Ax = 0.
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摘要：To account for row exchanges in Gaussian elimination, we include a permutation matrix P in the factorization PA = LU. Then we learn about vector spaces and subspaces; these are central to linear algebra.
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摘要：This session explains inverses, transposes and permutation matrices. We also learn how elimination leads to a useful factorization A = LU and how hard a computer will work to invert a very large matrix.
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摘要：This lecture looks at matrix multiplication from five different points of view. We then learn how to find the inverse of a matrix using elimination, and why the Gauss-Jordan method works.
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摘要：This session introduces the method of elimination, an essential tool for working with matrices. The method follows a simple algorithm. To help make sense of material presented later, we describe this algorithm in terms of matrix multiplication.
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摘要：A major application of linear algebra is to solving systems of linear equations. This lecture presents three ways of thinking about these systems. The "row method" focuses on the individual equations, the "column method" focuses on combining the columns, and the "matrix method" is an even more compact and powerful way of describing systems of linear equations.
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