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 ? so that the functions ?(?) = ? and
?(?) = ?? + 2 are orthogonal relative to the given inner
product.
(A.5) Consider the subspace ℙ1 of ?[0,2] consisting of all polynomials of degree 1 or less. Use the Gram-Schmidt process to convert its standard basis ? = {1,?} to an orthonormal basis.
(A.6) Obtain the polynomial ?(?) ∈ ℙ1 that “best approximates” ?(?) = ?2 in the sense that ?(?(?),?(?)) is minimized. Express your final answer in the form ?(?) = ?? + ? where ? and ? are simplified as much as possible (no decimal approximations).
Homework Answers
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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:...
[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.1) 〈?,1〉 (A.2) ‖ ? − 1 ‖ (A.3) ?(??,?? + 10), i.e. the distance between ?? and ?? +...
[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.1) 〈?,1〉 (A.2) ‖ ? − 1 ‖ (A.3) ?(??,?? + 10), i.e. the distance between ?? and ?? +...
[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.1) 〈?,1〉 (A.2) ‖ ? − 1 ‖ (A.3) ?(??,?? + 10), i.e. the distance between ?? and ?? +...
4. Consider R2x2 with inner product (A, B) tr(ATB), and let V CR2x2 be the subspace...
4. Consider R2x2 with inner product (A, B) tr(ATB), and let V CR2x2 be the subspace 1 1 1 0 This is consisting of upper-triangular matrices. Let B= a basis for V. (You do not need to prove this.) (a) (8 points) Use the Gram-Schmidt procedure on 3 to find an orthonormal basis for V. Find projy (B) (b) (4 points) Let B=
4. Consider R2x2 with inner product (A, B) tr(ATB), and let V CR2x2 be the subspace 1...
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),...
(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...
6. (15 pts) Consider an inner product on the vector space P2[-1, 1] of polynomials of...
6. (15 pts) Consider an inner product on the vector space P2[-1, 1] of polynomials of degree 2 or less in the closed interval [-1, 1], defined as follows: (f, 9) = | f(t)g(t) dt, for all f, ge P2[-1, 1]. Apply the Gram-Schmidt process to the basis {3, t – 2,t2 + 1} to obtain an {x1, X2, X3} = %3D orthonormal basis.
4) Consider the inner product space P2(R), with inner product 0 (a) Use the Gram-Schmidt process...
4) Consider the inner product space P2(R), with inner product 0 (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: oe d(an!)where a 1, anor n1,2....) ane- n!), where do -I, ln
5. Consider the following basis for P: B = {1,3+20+2=+2x"} Consider the inner product (p, q)...
5. Consider the following basis for P: B = {1,3+20+2=+2x"} Consider the inner product (p, q) = Sp(s)q()dr. Alternatively, (r", ) = Tiltz. (a) Use the Gram-Schmidt orthogonalization process to replace B with an orthogonal basis D. (b) Is D an orthonormal basis? Why or why not?
)-(Au) (Av), where A 10 3. Consider IR3 endowed with the inner product (u, v (a) Apply the Gram-Schmidt algorithm to th...
)-(Au) (Av), where A 10 3. Consider IR3 endowed with the inner product (u, v (a) Apply the Gram-Schmidt algorithm to the standard basis to obtain an orthonormal basis B. (b) Let v (1,-1,-2). Express v as linear combination of the elements of the orthonormal basis found in Part (a)
)-(Au) (Av), where A 10 3. Consider IR3 endowed with the inner product (u, v (a) Apply the Gram-Schmidt algorithm to the standard basis to obtain an orthonormal basis B....
4. Consider R2x2 with inner product (A, B) tr(ATB), and let V CR2x2 be the subspace 1 1 1 0 This is consisting of upper-triangular matrices. Let B= a basis for V. (You do not need to prove this.) (a) (8 points) Use the Gram-Schmidt procedure on 3 to find an orthonormal basis for V. Find projy (B) (b) (4 points) Let B=
4. Consider R2x2 with inner product (A, B) tr(ATB), and let V CR2x2 be the subspace 1...
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),...
(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...
6. (15 pts) Consider an inner product on the vector space P2[-1, 1] of polynomials of degree 2 or less in the closed interval [-1, 1], defined as follows: (f, 9) = | f(t)g(t) dt, for all f, ge P2[-1, 1]. Apply the Gram-Schmidt process to the basis {3, t – 2,t2 + 1} to obtain an {x1, X2, X3} = %3D orthonormal basis.
4) Consider the inner product space P2(R), with inner product 0 (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: oe d(an!)where a 1, anor n1,2....) ane- n!), where do -I, ln
5. Consider the following basis for P: B = {1,3+20+2=+2x"} Consider the inner product (p, q) = Sp(s)q()dr. Alternatively, (r", ) = Tiltz. (a) Use the Gram-Schmidt orthogonalization process to replace B with an orthogonal basis D. (b) Is D an orthonormal basis? Why or why not?
)-(Au) (Av), where A 10 3. Consider IR3 endowed with the inner product (u, v (a) Apply the Gram-Schmidt algorithm to the standard basis to obtain an orthonormal basis B. (b) Let v (1,-1,-2). Express v as linear combination of the elements of the orthonormal basis found in Part (a)
)-(Au) (Av), where A 10 3. Consider IR3 endowed with the inner product (u, v (a) Apply the Gram-Schmidt algorithm to the standard basis to obtain an orthonormal basis B....
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