Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Please do all the parts and please explain.
2. L2 and L1 in 3 dimensions The "unit ball" in L2 in 3 dimensions is easy to visualize-it's just a sphere of radius 1. The "unit ball" in L1 is a little more complicated. A. The "unit ball" in L1 is the set of points in (x₁, x2, x3) such that |x₁| + |x₂| + |x3| = 1. Describe the shape of this "unit ball" (Hint: it's a polyhedron. What's the name of the polyhedron? How many edges, vertices, and faces?) B. Consider the family of functions haα₂₁α3 (X₁, X₂, X3) = (x₁ − α₁)² + (x₂ − α₂)² + (x3 - α3)². For every positive real number r, define the set A₁azar as follows: A₁,92,93,r = {(X₁, X2, X3) € R³ such that (ha,2,3 (x₁, x₂, x3) — µ²)² is minimized. Give a geometric description of the set A₁,2,3, (i.e. what is the shape? How big is it? Where is it located?) C. Give a geometric description of the point on the set Aa₁a₂a3r that is closest to the origin. Illustrate your description with a sketch using the set A1,1,1,1 D. In your sketch, show also the unique ball in L2 centered at (0,0,0) that touches the set A1,1,1,1 at exactly one point. E. What is the relation between your answers to C and D?
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Transcribed Image Text:2. L2 and L1 in 3 dimensions The "unit ball" in L2 in 3 dimensions is easy to visualize-it's just a sphere of radius 1. The "unit ball" in L1 is a little more complicated. A. The "unit ball" in L1 is the set of points in (x₁, x2, x3) such that |x₁| + |x₂| + |x3| = 1. Describe the shape of this "unit ball" (Hint: it's a polyhedron. What's the name of the polyhedron? How many edges, vertices, and faces?) B. Consider the family of functions haα₂₁α3 (X₁, X₂, X3) = (x₁ − α₁)² + (x₂ − α₂)² + (x3 - α3)². For every positive real number r, define the set A₁azar as follows: A₁,92,93,r = {(X₁, X2, X3) € R³ such that (ha,2,3 (x₁, x₂, x3) — µ²)² is minimized. Give a geometric description of the set A₁,2,3, (i.e. what is the shape? How big is it? Where is it located?) C. Give a geometric description of the point on the set Aa₁a₂a3r that is closest to the origin. Illustrate your description with a sketch using the set A1,1,1,1 D. In your sketch, show also the unique ball in L2 centered at (0,0,0) that touches the set A1,1,1,1 at exactly one point. E. What is the relation between your answers to C and D?
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