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Unformatted text preview: L. Vandenberghe 12/10/09 EE103 Final Exam You have time until 11:00AM. Only this booklet should be on your desk. Calculators are not allowed. Please turn off and put away your cellphones. Write your answers neatly and concisely in the space provided after each question. Provide enough detail to convince us that you derived, not guessed, your answers. Name: Student ID#: Your left neighbors name: Your right neighbors name: Problem 1 /10 Problem 2 /10 Problem 3 /10 Problem 4 /10 Problem 5 /10 Problem 6 /10 Problem 7 /10 Total /70 Important formulas Vectors and matrices Relation between inner product and angle: x T y = bardbl x bardblbardbl y bardbl cos negationslash ( x,y ) Flop counts for basic operations ( is a scalar, x and y are n-vectors, A is an m n-matrix, B is an n p-matrix) Inner product x T y : 2 n 1 2 n for large n Vector addition x + y : n Scalar multiplication x : n Matrix-vector product Ax : m (2 n 1) 2 mn for large n Matrix-matrix product AB : mp (2 n 1) 2 mpn for large n Solving linear equations Cost of solving Ax = b when A is n n and upper or lower triangular: n 2 flops Cost of Cholesky factorization A = LL T : (1 / 3) n 3 flops if A is n n Cost of LU factorization A = PLU : (2 / 3) n 3 flops if A is n n Matrix norm and condition number Definition of matrix norm: bardbl A bardbl = max x negationslash =0 bardbl Ax bardbl bardbl x bardbl Properties of the matrix norm bardbl A bardbl = max bardbl x bardbl =1 bardbl Ax bardbl bardbl A bardbl = | |bardbl A bardbl for R bardbl A bardbl 0 for all A ; bardbl A bardbl = 0 only if A = 0 bardbl A + B bardbl bardbl A bardbl + bardbl B bardbl bardbl A bardbl = bardbl A T bardbl bardbl Ax bardbl bardbl A bardblbardbl x bardbl if the matrix-vector product exists bardbl AB bardbl bardbl A bardblbardbl B bardbl if the matrix product exists bardbl A bardblbardbl A 1 bardbl 1 if A is square and nonsingular...
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- Spring '08