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3. Consider the vector space of real-valued, continuous functions defined over [1,2], which we denote by C[1,2]. We equip C[1,2] with the L² norm, 1/2 ||S|| = (S{* (S(x))² dr), vƒ € C[1,2]. Find pЄ P₁ closest to f(x) = 1/x in the L² norm over the interval [1,2]. In other words, find pЄ Pi that minimizes ||p- ƒ ||². Hint for problem 3: Assume p (z) = a+b. The goal is to determine a and b by minimizing p(a, b) = ||az + b = f². The minimizer occurs when the partial derivatives are zero, ie, solve for (a, b) such that 이(a,b) = 0 and (1,6) = 0.

3. Consider the vector space of real-valued, continuous functions defined over [1,2], which we denote by C[1,2]. We equip C[1,2] with the L² norm, 1/2 ||S|| = (S{* (S(x))² dr), vƒ € C[1,2]. Find pЄ P₁ closest to f(x) = 1/x in the L² norm over the interval [1,2]. In other words, find pЄ Pi that minimizes ||p- ƒ ||². Hint for problem 3: Assume p (z) = a+b. The goal is to determine a and b by minimizing p(a, b) = ||az + b = f². The minimizer occurs when the partial derivatives are zero, ie, solve for (a, b) such that 이(a,b) = 0 and (1,6) = 0.

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter4: Vector Spaces
Section4.4: Spanning Sets And Linear Independence
Problem 76E: Let f1(x)=3x and f2(x)=|x|. Graph both functions on the interval 2x2. Show that these functions are...
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3. Consider the vector space of real-valued, continuous functions defined over [1,2], which we denote by C[1,2]. We equip C[1,2] with the L² norm, 1/2 ||S|| = (S{* (S(x))² dr), vƒ € C[1,2]. Find pЄ P₁ closest to f(x) = 1/x in the L² norm over the interval [1,2]. In other words, find pЄ Pi that minimizes ||p- ƒ ||². Hint for problem 3: Assume p (z) = a+b. The goal is to determine a and b by minimizing p(a, b) = ||az + b = f². The minimizer occurs when the partial derivatives are zero, ie, solve for (a, b) such that 이(a,b) = 0 and (1,6) = 0.
Transcribed Image Text:3. Consider the vector space of real-valued, continuous functions defined over [1,2], which we denote by C[1,2]. We equip C[1,2] with the L² norm, 1/2 ||S|| = (S{* (S(x))² dr), vƒ € C[1,2]. Find pЄ P₁ closest to f(x) = 1/x in the L² norm over the interval [1,2]. In other words, find pЄ Pi that minimizes ||p- ƒ ||². Hint for problem 3: Assume p (z) = a+b. The goal is to determine a and b by minimizing p(a, b) = ||az + b = f². The minimizer occurs when the partial derivatives are zero, ie, solve for (a, b) such that 이(a,b) = 0 and (1,6) = 0.
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