C. There is a geometric way to represent the solution that you found in (B). We will show this in stages. i. ii. iv. The set of points in the (x₁, x₂) plane that satisfies |x₁|² + 1x₂|² = 1 is called the unit ball in L2(x₁, x₂). Draw this unit ball in figure on the graph you drew in (A) above. The set of points in the (x₁, x₂)plane that satisfies |x₁|² + 1x₂1² = r² is called the ball of radius r in L2(x₁, x₂ On the same graph as (A) above, draw the L2 balls of radius 1,2,3. What is the geometrical relationship between your answer to C.ii. and your answer to B above? Give a mathematical explanation of your answer to C.iii.
C. There is a geometric way to represent the solution that you found in (B). We will show this in stages. i. ii. iv. The set of points in the (x₁, x₂) plane that satisfies |x₁|² + 1x₂|² = 1 is called the unit ball in L2(x₁, x₂). Draw this unit ball in figure on the graph you drew in (A) above. The set of points in the (x₁, x₂)plane that satisfies |x₁|² + 1x₂1² = r² is called the ball of radius r in L2(x₁, x₂ On the same graph as (A) above, draw the L2 balls of radius 1,2,3. What is the geometrical relationship between your answer to C.ii. and your answer to B above? Give a mathematical explanation of your answer to C.iii.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Please do part C and all the parts and explain.
Transcribed Image Text:1. Comparing L2 and L1
A. Let S be the set of points in the (x₁, x₂) plane that minimizes the function f(x₁, x₂) =
(4x₁ + 3x₂ - 10)². Describe Sf using set notation, and draw a graph in the
(x₁, x₂) plane that shows the set Sf (Hint: the set is a line).
B. From (A) we know there is more than one pair (x₁, x₂) that minimizes f(x₁,x₂). Find
the point (x₁, x₂) in Sƒ that minimizes the length of the vector (x₁, x₂) (note the length
of the vector is the distance between (0,0) and the point (x₁, x₂).
C. There is a geometric way to represent the solution that you found in (B). We will
show this in stages.
i.
ii.
iv.
The set of points in the (x₁, x₂) plane that satisfies |x₁|²2 +|x₂|² = 1 is called
the unit ball in L2(x₁, x₂). Draw this unit ball in figure on the graph you drew in
(A) above.
The set of points in the (x₁, x₂)plane that satisfies |x₁|² + 1x₂1² = r² is called
the ball of radius r in L2(x₁, x₂ On the same graph as (A) above, draw the L2
balls of radius 12,2,3.
What is the geometrical relationship between your answer to C.ii. and your
answer to B above?
Give a mathematical explanation of your answer to C.iii.
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