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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 neighbor’s name: Your right neighbor’s 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 ( α 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 αx : 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 αA bardbl = | α |bardbl A bardbl for α ∈ 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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