![2. Consider the vector space C([0, 1]) consisting of all continuous functions f: [0,1]-R with the weighted inner product, (f.](http://img.homeworklib.com/images/7f7fad33-86bf-4b27-946b-6f8071a2acfd.png?x-oss-process=image/resize,w_560)
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2. Consider the vector space C([0, 1]) consisting of all continuous functions f: [0,1]-R with the...
2. (a) Prove that 1 (f, g)=| x2 f(x)g(x)dx is an inner product on the vector space C(I-1,1) of co...
NEED (B) AND (C)
2. (a) Prove that 1 (f, g)=| x2 f(x)g(x)dx is an inner product on the vector space C(I-1,1) of continuous real-valued funo- tions on the domain [-1, 1] (b) Use the Gram-Schmidt process to find an orthonormal basis for P2(R) with re- spect to this inner product (c) Find a polynomial q(x) such that for every p E P2R
2. (a) Prove that 1 (f, g)=| x2 f(x)g(x)dx is an inner product on the vector space...
Let clo, π] := {f : [0, π] → R I f is continuous). With addition and scalar multiplication defined in the usual way, th...
Let clo, π] := {f : [0, π] → R I f is continuous). With addition and scalar multiplication defined in the usual way, this is a vector space. Let the inner product on CO,T] be defined analogous to (21), that is, (me) :-o u(z)r(z) dz. sinx and g(x) = 2.2. Which is "bigger": f or g? (a) Let f(x) (b) g? xplain. (c) Find a nontrivial function in CIO, π], which is orthogonal to f. d) Find a nontrivial...
5. (15') Define the inner-product on C([-1,1]), the space of all continuous functions on the interval...
5. (15') Define the inner-product on C([-1,1]), the space of all continuous functions on the interval [-1, 1), by (5(2), gla) - s(z)g(z) dr. (a) Use Gram-Schmidt algorithm to convert the set (1,1 + 1,(1 + x)2} to an orthogonal set. (b) Is the set you found in Part (a) still orthogonal if the interval of integral in the definition of inner-product is changed to [0, 1]? Explain your answer.
Advanced linear algebra thxxxxxxxx Consider the complex vector space P4(C) of polynomials of deg...
advanced linear algebra thxxxxxxxx
Consider the complex vector space P4(C) of polynomials of
degree at most 4 with coeffi- cients in C, equipped with the inner
product ⟨ , ⟩ defined by
5. Consider the complex vector space P4(C) of polynomials of degree at most 4 with coeffi- cients in C, equipped with the inner product (, ) defined by (f, g)fx)g(xJdx. (a) Find an orthogonal basis of the subspace Pi(C)span,x (b) Find the element of Pi (C) that is...
5. (15') Define the inner-product on C([-1,1]), the space of all continuous functions on the interval...
5. (15') Define the inner-product on C([-1,1]), the space of all continuous functions on the interval [-1,1], by (f(a), g(x) = $ $(a)g(x) dr. (a) Use Gram-Schmidt algorithm to convert the set {1,1 + ,(1+x)?} to an orthogonal set. (b) Is the set you found in Part (a) still orthogonal if the interval of integral in the definition of inner-product is changed to [0, 1]? Explain your an answer.
- Let V be the vector space of continuous functions defined f : [0,1] → R...
- Let V be the vector space of continuous functions defined f : [0,1] → R and a : [0, 1] →R a positive continuous function. Let < f, g >a= Soa(x)f(x)g(x)dx. a) Prove that <, >a defines an inner product in V. b) For f,gE V let < f,g >= So f(x)g(x)dx. Prove that {xn} is a Cauchy sequence in the metric defined by <, >a if and only if it a Cauchy sequence in the metric defined by...
4) Consider the inner product space P2(R), with inner product (a) Use the Gram-Schmidt process to...
4) Consider the inner product space P2(R), with inner product (a) Use the Gram-Schmidt process to construct an orthonormal basis from the basis (b) Using your answer to part (a), give the least squares approximation in P2(R) to the function f(x)on the interval [0, 1. Hint: You may use the following result without proof f Ine* dr = (-1)"(ane-n!), where ao = 1, an- | n. + | , for n-1, 2, ).
4) Consider the inner product space P2(R),...
A] Consider the inner product space obtained by equipping ?[0,2] with the inner product given below:...
A] Consider the inner product space obtained by equipping ?[0,2] with the inner product given below: 〈?(?),?(?)〉 = ∫ ?(?)?(?)?? 2 0 Determine the value of each of the following (simplifying where possible; no decimals). You do not have to show the steps of calculating the integrals, but must at least write the integrals used in your calculations. (A.3) ?(??,?? + 10), i.e. the distance between ?? and ?? + 10 . (A.4) Determine the value of ?...
4. (6pt) Use the inner product (f,g)f ds to determine the following. (a) Determine if the functio...
i need help with this linear Algebra question
4. (6pt) Use the inner product (f,g)f ds to determine the following. (a) Determine if the function g(z) = z2-3x + 2 or h(x) = x2-2x + 1 is closest to the fl () is closest to the function f)2+2 on (b, Show that (1,2r - 1) is an orthogonal set (c) Beginning with the basis (1,2 1, 2 (d) Find an orthonormal basis for P2. (e) Find the least squares quadratic...
(4) Consider the inner product space P2(R), with inner product (a) Use the Gram-Schmidt process t...
(4) Consider the inner product space P2(R), with inner product (a) Use the Gram-Schmidt process to construct an orthonormal basis from the basis 11, r, r2) b) Using your answer to part (a), give the least squares approximation in P2(R) to the function f(x) e on the interval [0, Hint: You may use the following result without proof: J İlne dra(-1)"(ane-n!), where ao-1, an-le! + îl , for n-1, 2, or n=1,2 .. ).
(4) Consider the inner product space...
NEED (B) AND (C)
2. (a) Prove that 1 (f, g)=| x2 f(x)g(x)dx is an inner product on the vector space C(I-1,1) of continuous real-valued funo- tions on the domain [-1, 1] (b) Use the Gram-Schmidt process to find an orthonormal basis for P2(R) with re- spect to this inner product (c) Find a polynomial q(x) such that for every p E P2R
2. (a) Prove that 1 (f, g)=| x2 f(x)g(x)dx is an inner product on the vector space...
Let clo, π] := {f : [0, π] → R I f is continuous). With addition and scalar multiplication defined in the usual way, this is a vector space. Let the inner product on CO,T] be defined analogous to (21), that is, (me) :-o u(z)r(z) dz. sinx and g(x) = 2.2. Which is "bigger": f or g? (a) Let f(x) (b) g? xplain. (c) Find a nontrivial function in CIO, π], which is orthogonal to f. d) Find a nontrivial...
5. (15') Define the inner-product on C([-1,1]), the space of all continuous functions on the interval [-1, 1), by (5(2), gla) - s(z)g(z) dr. (a) Use Gram-Schmidt algorithm to convert the set (1,1 + 1,(1 + x)2} to an orthogonal set. (b) Is the set you found in Part (a) still orthogonal if the interval of integral in the definition of inner-product is changed to [0, 1]? Explain your answer.
advanced linear algebra thxxxxxxxx
Consider the complex vector space P4(C) of polynomials of
degree at most 4 with coeffi- cients in C, equipped with the inner
product ⟨ , ⟩ defined by
5. Consider the complex vector space P4(C) of polynomials of degree at most 4 with coeffi- cients in C, equipped with the inner product (, ) defined by (f, g)fx)g(xJdx. (a) Find an orthogonal basis of the subspace Pi(C)span,x (b) Find the element of Pi (C) that is...
5. (15') Define the inner-product on C([-1,1]), the space of all continuous functions on the interval [-1,1], by (f(a), g(x) = $ $(a)g(x) dr. (a) Use Gram-Schmidt algorithm to convert the set {1,1 + ,(1+x)?} to an orthogonal set. (b) Is the set you found in Part (a) still orthogonal if the interval of integral in the definition of inner-product is changed to [0, 1]? Explain your an answer.
- Let V be the vector space of continuous functions defined f : [0,1] → R and a : [0, 1] →R a positive continuous function. Let < f, g >a= Soa(x)f(x)g(x)dx. a) Prove that <, >a defines an inner product in V. b) For f,gE V let < f,g >= So f(x)g(x)dx. Prove that {xn} is a Cauchy sequence in the metric defined by <, >a if and only if it a Cauchy sequence in the metric defined by...
4) Consider the inner product space P2(R), with inner product (a) Use the Gram-Schmidt process to construct an orthonormal basis from the basis (b) Using your answer to part (a), give the least squares approximation in P2(R) to the function f(x)on the interval [0, 1. Hint: You may use the following result without proof f Ine* dr = (-1)"(ane-n!), where ao = 1, an- | n. + | , for n-1, 2, ).
4) Consider the inner product space P2(R),...
i need help with this linear Algebra question
4. (6pt) Use the inner product (f,g)f ds to determine the following. (a) Determine if the function g(z) = z2-3x + 2 or h(x) = x2-2x + 1 is closest to the fl () is closest to the function f)2+2 on (b, Show that (1,2r - 1) is an orthogonal set (c) Beginning with the basis (1,2 1, 2 (d) Find an orthonormal basis for P2. (e) Find the least squares quadratic...
(4) Consider the inner product space P2(R), with inner product (a) Use the Gram-Schmidt process to construct an orthonormal basis from the basis 11, r, r2) b) Using your answer to part (a), give the least squares approximation in P2(R) to the function f(x) e on the interval [0, Hint: You may use the following result without proof: J İlne dra(-1)"(ane-n!), where ao-1, an-le! + îl , for n-1, 2, or n=1,2 .. ).
(4) Consider the inner product space...
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