Homework_2

Homework_2 - CSC 302 — Introduction to Numerical Methods...

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Unformatted text preview: CSC 302 — Introduction to Numerical Methods Homework #2 Question 1. Compute the determinants of the following matrices. A = 1 2 3 1 0 1 5 0 1 , B = 1- λ 2 3 1- λ 1 5 1- λ Answer: For matrix A it is easiest to expand along the second column. We obtain det( A ) = a 12 × cofactor ( A 12 ) = 2 × (- 1) 3 1 1 5 1 = 2 × (- 1) 3 × (- 4) = 8 . We shall also expand B along its second column. det( B ) = b 12 × cofactor ( B 12 ) + b 22 × cofactor ( B 22 ) = 2 × (- 1) 3 1 1 5 1- λ + (- λ ) × (- 1) 4 1- λ 3 5 1- λ = 2 × (- 1) 3 [(1- λ )- 5] + (- λ ) × (- 1) 4 [(1- λ )(1- λ )- 15] = 2( λ + 4)- λ λ 2- 2 λ- 14 =- λ 3 + 2 λ 2 + 16 λ + 8 . 1 Question 2. Let x ∈ < n and consider the function f ( x ) = α k x k 1 + β k x k ∞ where α and β are two nonzero scalars. Check this function against the three conditions necessary for it to be a vector norm and determine if in fact it is a norm. Answer: (a) Definite/Positivity: f ( x ) = α || x || 1 + β || x || ∞ is not necessarily ≥ For example, α =- 1, β =- 1, then f ( x ) < 0. So, fails positivity: will not be a norm. (b) Homogeneity: f ( δx ) = α || δx || 1 + β || δx || ∞ = α | δ ||| x || 1 + β | δ ||| x || ∞ = | δ | ( α || x || 1 + β || x || ∞ ) = | δ | · f ( x ) So, it passes homogeneity. (c) Triangle inequality: f ( x + y ) = α || x + y || 1 + β || x + y || ∞ ≤ α ( || x || 1 + || y || 1 ) + β ( || x || ∞ + || y || ∞ ) f ( x + y ) ≤ ( α || x || 1 + β || x || ∞ ) + ( α || y || 1 + β || y || ∞ ) f ( x + y ) ≤ f ( x ) + f ( y ) So, it passes triangle inequality....
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This note was uploaded on 02/16/2011 for the course CS 302 taught by Professor Stewart during the Spring '11 term at N.C. State.

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Homework_2 - CSC 302 — Introduction to Numerical Methods...

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