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3320hw3sol

# 3320hw3sol - Math 3320 Problem Set 3 Solutions 1 1 This...

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Unformatted text preview: Math 3320 Problem Set 3 Solutions 1 1. This exercise is intended to illustrate the proof of the Theorem on page 15 of Chapter 1 in the concrete case of the continued fraction 1 upslope ր 2 + 1 upslope ր 3 + 1 upslope ր 4 + 1 upslope ր 5 . (a) Write down the product A 1 A 2 A 3 A 4 = parenleftBigg 1 1 a 1 parenrightBiggparenleftBigg 1 1 a 2 parenrightBiggparenleftBigg 1 1 a 3 parenrightBiggparenleftBigg 1 1 a 4 parenrightBigg as- sociated to 1 upslope ր 2 + 1 upslope ր 3 + 1 upslope ր 4 + 1 upslope ր 5 . Solution : A 1 A 2 A 3 A 4 = parenleftBig 0 1 1 2 parenrightBigparenleftBig 0 1 1 3 parenrightBigparenleftBig 0 1 1 4 parenrightBigparenleftBig 0 1 1 5 parenrightBig (b) Compute the four matrices A 1 , A 1 A 2 , A 1 A 2 A 3 , A 1 A 2 A 3 A 4 and relate these to the edges of the zigzag path in the strip of triangles for 1 upslope ր 2 + 1 upslope ր 3 + 1 upslope ր 4 + 1 upslope ր 5 . Solution : A 1 = parenleftBig 0 1 1 2 parenrightBig , A 1 A 2 = parenleftBig 0 1 1 2 parenrightBigparenleftBig 0 1 1 3 parenrightBig = parenleftBig 1 3 2 7 parenrightBig , A 1 A 2 A 3 = parenleftBig 1 3 2 7 parenrightBigparenleftBig 0 1 1 4 parenrightBig = parenleftBig 3 13 7 30 parenrightBig , A 1 A 2 A 3 A 4 = parenleftBig 3 13 7 30 parenrightBigparenleftBig 0 1 1 5 parenrightBig = parenleftBig 13 68 30 157 parenrightBig . These four matrices, together with the identity matrix parenleftBig 1 0 0 1 parenrightBig , correspond to the edges of the zigzag path, with labels given by the columns of the matrices. (c) Compute the four matrices A 4 , A 3 A 4 , A 2 A 3 A 4 , A 1 A 2 A 3 A 4 and relate these to the successive fractions that one gets when one computes the value of 1 upslope ր 2 + 1 upslope ր 3 + 1 upslope ր 4 + 1 upslope ր 5 , namely 1 upslope ր 5 , 1 upslope ր 4 + 1 upslope ր 5 , 1 upslope ր 3 + 1 upslope ր 4 + 1 upslope ր 5 , and 1 upslope ր 2 + 1 upslope ր 3 + 1 upslope ր 4 + 1 upslope ր 5 . Solution : A 4 = parenleftBig 0 1 1 5 parenrightBig , A 3 A 4 = parenleftBig 0 1 1 4 parenrightBigparenleftBig 0 1 1 5 parenrightBig = parenleftBig 1 5 4 21 parenrightBig , A 2 A 3 A 4 = parenleftBig 0 1 1 3 parenrightBigparenleftBig 1 5 4 21 parenrightBig = parenleftBig 4 21 13 68 parenrightBig , A 1 A 2 A 3 A 4 = parenleftBig...
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3320hw3sol - Math 3320 Problem Set 3 Solutions 1 1 This...

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