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Unformatted text preview: ENEE 241 02 * READING ASSIGNMENT 8 Thu 2/26 Lecture Topics: Gaussian elimination, continued; row interchanges; singularity; linear independence Textbook References: sections 2.7, 2.8, 2.4, 2.5 Key Points: • In Gaussian elimination, the row multipliers m ij and the upper-triangular matrix U can be computed offline, i.e., using forward elimination on the matrix A before the vector b is given. • Zero pivots can be dealt with using row interchanges (row pivoting). • If all unused rows have zero entries in the pivot column, the corresponding diagonal entry in U is zero and the matrix A is singular. • If A is singular, the equation Ax = b has a solution for certain values of b only, and that solution is not unique. • Nonsingularity of A is equivalent to linear independence of its columns, i.e., the equation Ax = having x = as its unique solution. • More generally, a set of vectors is linearly independent if the only linear combination (of these vectors) that produces the all-zeros vector has all its coefficients equal to zero. Theory and Examples: 1 . We saw that forward elimination reduces the system of simultaneous equations Ax = b to an upper triangular system Ux = c , which is then solved using backward substitution. 2 . The row multipliers m ij used in forward elimination, as well as the upper-triangular matrix U obtained at the end of this procedure, are determined by the matrix A alone; they do not depend on the r.h.s. vector b . Thus a large part of the computation can be carried out without knowing b . Once b is given, the multipliers m ij can be used to obtain the vector...
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- Spring '08
- Linear Algebra, ax, forward elimination