linear_systems

linear_systems - Chapter 2 Linear Systems of Equations •...

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Unformatted text preview: Chapter 2 Linear Systems of Equations • Existence, uniqueness • Conditioning (sensitivity of solution) • If A is nonsingular, inverse A-1 exists • x=A-1 b Gaussian Elimination with Partial Pivoting 1 3 2 7 Ρ Σ ΢ Τ Φ Υ x 1 x 2 Ρ Σ ΢ Τ Φ Υ = b 1 b 2 Ρ Σ ΢ Τ Φ Υ ⇔ 1 3 1 Ρ Σ ΢ Τ Φ Υ x 1 x 2 Ρ Σ ΢ Τ Φ Υ = b 1 b 2 − 2 b 1 Ρ Σ ΢ Τ Φ Υ 10 − 6 3 1 7 Ρ Σ ΢ Τ Φ Υ x 1 x 2 Ρ Σ ΢ Τ Φ Υ = b 1 b 2 Ρ Σ ΢ Τ Φ Υ ⇔ 10 − 6 3 7 − 3 x 10 6 Ρ Σ ΢ Τ Φ Υ x 1 x 2 Ρ Σ ΢ Τ Φ Υ = b 1 b 2 − 10 6 b 1 Ρ Σ ΢ Τ Φ Υ Generated a very large number Partial pivoting: use the largest number in the column (in absolute value) Nonsigularity and Singularity • The (square) n x n matrix A is nonsingular if: • Inverse A-1 exists • det(A) is nonzero • rank(A)=n (rank: no. of linearly independent rows) • for any A is nonsingular Ax=b has a unique solution for any b The singular case • If A is singular (inverse does not exist) • Ax=b has – An infinite number of solutions if b is in the span(A) – No solutions otherwise Example: Known Material...
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This note was uploaded on 01/15/2012 for the course ECON 101 taught by Professor N/a during the Fall '11 term at Middlesex CC.

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linear_systems - Chapter 2 Linear Systems of Equations •...

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