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Unformatted text preview: CMPSC/MATH 451 Numerical Computations Lecture 6 September 2, 2011 Prof. Kamesh Madduri REVIEW 2 Systems of Linear Equations: Preliminaries • Linearlyindependent vectors • Rank of a matrix • Span(A) • Ax=b • When is the matrix A nonsingular? • Matrix singularity tells us whether the solution exists and is unique or not • Vector norms • Relationship between 1norm, 2norm and Inftynorm 3 Systems of Linear Equations • Covered on blackboard, corresponding slides from textbook follow. 4 Existence, Uniqueness, and Conditioning Solving Linear Systems Special Types of Linear Systems Software for Linear Systems Singularity and Nonsingularity Norms Condition Number Error Bounds Properties of Vector Norms For any vector norm k x k > if x 6 = k γ x k =  γ  · k x k for any scalar γ k x + y k ≤ k x k + k y k (triangle inequality) In more general treatment, these properties taken as definition of vector norm Useful variation on triangle inequality k x k  k y k ≤ k x y k Michael T. Heath Scientific Computing 12 / 88 Existence, Uniqueness, and Conditioning Solving Linear Systems Special Types of Linear Systems Software for Linear Systems Singularity and Nonsingularity Norms Condition Number Error Bounds Matrix Norms Matrix norm corresponding to given vector norm is defined by k A k = max x 6 = k Ax k k x k Norm of matrix measures maximum stretching matrix does to any vector in given vector norm Michael T. Heath Scientific Computing 13 / 88 Existence, Uniqueness, and Conditioning Solving Linear Systems Special Types of Linear Systems Software for Linear Systems Singularity and Nonsingularity Norms Condition Number Error Bounds Matrix Norms Matrix norm corresponding to vector 1norm is maximum absolute column sum k A k 1 = max j n X i =1  a ij  Matrix norm corresponding to vector ∞norm is maximum absolute row sum k A k ∞ = max i n X j =1  a ij  Handy way to remember these is that matrix norms agree with corresponding vector norms for n × 1 matrix Michael T. Heath Scientific Computing 14 / 88 Existence, Uniqueness, and Conditioning Solving Linear Systems Special Types of Linear Systems Software for Linear Systems Singularity and Nonsingularity Norms Condition Number Error Bounds Properties of Matrix Norms Any matrix norm satisfies k A k > if A 6 = k γ A k =  γ  · k A k for any scalar γ k A + B k ≤ k A...
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