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Problem 1. Let the inner prodct )be deined by (u.v)xu (x) v (x) dx, and let the norm |I-ll be def...
Problem 1. Let the inner product (,) be defined by (u.v)xu (x)v (x) dx, and let the norm Iilbe de...
Problem 1. Let the inner product (,) be defined by (u.v)xu (x)v (x) dx, and let the norm Iilbe defined by lIul-)Corhe target funtio), and work with the approximating space P4 Use Gram-Schmidt orthogonalization with this inner product to find orthogonal polynomials (x) through degree four. Standardize your polynomials such that p: (1) 1. (a) Form the five-by-five Gram matrix for this inner product with the basis functions p (x) degree 4 approximation o f (x) using the specified norm,...
Problem 2. The Laguerre polynomials L (x) are orthogonal with respect to the inner product (u,u)=...
Problem 2. The Laguerre polynomials L (x) are orthogonal with respect to the inner product (u,u)=/o e-ru (x) u(x) dx and standardized so that L" (0) = 1 . In addition to their importance in numerical analysis, the La- guerre polynomials are notable for their use in electron orbitals in atoms Use Gram-Schmidt orthogonalization along with the standardization L(0)1 to find the Laguerre polynomials of degree ns4 The Laguerre polynomials obey the TTRR (n + 1) L" +1 (x) =...
Linear Algebra 2) General Inner Products, Length, Distance and Angle a) Determine if (u,v)-3uiv,-u,v, is a dot product b) Show that (u.v)-a+a,h,'2 is a product if a, 20 e)Let A-(41 ..)and B-G...
Linear Algebra
2) General Inner Products, Length, Distance and Angle a) Determine if (u,v)-3uiv,-u,v, is a dot product b) Show that (u.v)-a+a,h,'2 is a product if a, 20 e)Let A-(41 ..)and B-G ) Use inner product on 4 -2 M (A, B aitai +apb +2a to find the length of A, B, namely ll-41 and 1 d) Find the angle between the two matrices above e) Find the distance between the two above matrices 0) For the functions (x)-1 and...
6. Let p;(xi = 0,... , n}, with degp;(x) = i, be a set of orthogonal polynomials with respect to the inner product f f(...
6. Let p;(xi = 0,... , n}, with degp;(x) = i, be a set of orthogonal polynomials with respect to the inner product f f(x)g(x) dx. Given a < b, let q(x) be the line mapping a to -1 and b to 1. Prove {p;(q(x))|i = 0,... , n} is a set of orthogonal polynomials with respect to the inner product f(x)g(x) dz, satisfying deg p;(q(x))= i -
6. Let p;(xi = 0,... , n}, with degp;(x) = i, be...
1.(16) Let P be an inner product space with an inner product defined as <.g >...
1.(16) Let P be an inner product space with an inner product defined as <.g > Ox)g(x)dx a) Let / =1+x.8=-2+x-x. Compute: <.8 >. The angle between / and g, and proj, b) Let h=1+ mx' in P Find m such that and h are orthogonal c) Let B = (1+x.I-XX+X' is a basis for P. Use the Gram-Schmidt process to covert B to an orthogonal basis for P. 2. Suppose and ware vectors in an inner product space V...
let P3 denote the vector space of polynomials of degree 3 or less, with an inner product defined by 14. Let Ps denote the vector space of polynomials of degree 3 or less, with an inner product def...
let P3 denote the vector space of polynomials of degree 3 or
less, with an inner product defined by
14. Let Ps denote the vector space of polynomials of degree 3 or less, with an inner product defined by (p, q) Ji p(x)q(x) dr. Find an orthogo- nal basis for Ps that contains the vector 1+r. Find the norm (length) of each of your basis elements
14. Let Ps denote the vector space of polynomials of degree 3 or less,...
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 P3 have the inner product given by evaluation at -3, -1, 1, and 3....
== Let P3 have the inner product given by evaluation at -3, -1, 1, and 3. Let po(t) = 4, p1(t)=t, and t² – 5 q(t) = Notice that these polynomials form an orthogonal set with this inner product. Find the best 4 approximation to p(t) = tº by polynomials in Span{P0,21,9}. The best approximation to p(t) = tº by polynomials in Span{Po.21,93 is
QUESTION 1 Let V-L2([0,1 ],C) and > : Vx-СУч . Г f(x)g(x)dx be an inner product...
QUESTION 1 Let V-L2([0,1 ],C) and > : Vx-СУч . Г f(x)g(x)dx be an inner product on V Let gor 91, 92, 93:0,1]R be given by gox)-1,g1(x)-x, 920x)-x2, g3(x) -x3 and consider the following subset S = { go, g 1, g 2, g3JC V. After applying the Gram-Schmidt process the following set of vectors T = {vo, vľ, V2, V3} is an orthonormal set, where V1, V2, V3, and V4 are given by: O vo= 1, v,-V3(2x-1), v,-V5 (6x2-6x...
please provide step by step solution Thank you (1 point) Use the inner product (5,8) =...
please provide step by step solution
Thank you
(1 point) Use the inner product (5,8) = $* f()g(x) dx in the vector space P(R) of polynomials to find the orthogonal projection of f(x) = 2x2 + 4 onto the subspace V spanned by g(x) = x and h(x) = 1. (Caution: x and 1 do not form an orthogonal basis of V.) projy(f) =
Problem 1. Let the inner product (,) be defined by (u.v)xu (x)v (x) dx, and let the norm Iilbe defined by lIul-)Corhe target funtio), and work with the approximating space P4 Use Gram-Schmidt orthogonalization with this inner product to find orthogonal polynomials (x) through degree four. Standardize your polynomials such that p: (1) 1. (a) Form the five-by-five Gram matrix for this inner product with the basis functions p (x) degree 4 approximation o f (x) using the specified norm,...
Problem 2. The Laguerre polynomials L (x) are orthogonal with respect to the inner product (u,u)=/o e-ru (x) u(x) dx and standardized so that L" (0) = 1 . In addition to their importance in numerical analysis, the La- guerre polynomials are notable for their use in electron orbitals in atoms Use Gram-Schmidt orthogonalization along with the standardization L(0)1 to find the Laguerre polynomials of degree ns4 The Laguerre polynomials obey the TTRR (n + 1) L" +1 (x) =...
Linear Algebra
2) General Inner Products, Length, Distance and Angle a) Determine if (u,v)-3uiv,-u,v, is a dot product b) Show that (u.v)-a+a,h,'2 is a product if a, 20 e)Let A-(41 ..)and B-G ) Use inner product on 4 -2 M (A, B aitai +apb +2a to find the length of A, B, namely ll-41 and 1 d) Find the angle between the two matrices above e) Find the distance between the two above matrices 0) For the functions (x)-1 and...
6. Let p;(xi = 0,... , n}, with degp;(x) = i, be a set of orthogonal polynomials with respect to the inner product f f(x)g(x) dx. Given a < b, let q(x) be the line mapping a to -1 and b to 1. Prove {p;(q(x))|i = 0,... , n} is a set of orthogonal polynomials with respect to the inner product f(x)g(x) dz, satisfying deg p;(q(x))= i -
6. Let p;(xi = 0,... , n}, with degp;(x) = i, be...
1.(16) Let P be an inner product space with an inner product defined as <.g > Ox)g(x)dx a) Let / =1+x.8=-2+x-x. Compute: <.8 >. The angle between / and g, and proj, b) Let h=1+ mx' in P Find m such that and h are orthogonal c) Let B = (1+x.I-XX+X' is a basis for P. Use the Gram-Schmidt process to covert B to an orthogonal basis for P. 2. Suppose and ware vectors in an inner product space V...
let P3 denote the vector space of polynomials of degree 3 or
less, with an inner product defined by
14. Let Ps denote the vector space of polynomials of degree 3 or less, with an inner product defined by (p, q) Ji p(x)q(x) dr. Find an orthogo- nal basis for Ps that contains the vector 1+r. Find the norm (length) of each of your basis elements
14. Let Ps denote the vector space of polynomials of degree 3 or less,...
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 P3 have the inner product given by evaluation at -3, -1, 1, and 3. Let po(t) = 4, p1(t)=t, and t² – 5 q(t) = Notice that these polynomials form an orthogonal set with this inner product. Find the best 4 approximation to p(t) = tº by polynomials in Span{P0,21,9}. The best approximation to p(t) = tº by polynomials in Span{Po.21,93 is
QUESTION 1 Let V-L2([0,1 ],C) and > : Vx-СУч . Г f(x)g(x)dx be an inner product on V Let gor 91, 92, 93:0,1]R be given by gox)-1,g1(x)-x, 920x)-x2, g3(x) -x3 and consider the following subset S = { go, g 1, g 2, g3JC V. After applying the Gram-Schmidt process the following set of vectors T = {vo, vľ, V2, V3} is an orthonormal set, where V1, V2, V3, and V4 are given by: O vo= 1, v,-V3(2x-1), v,-V5 (6x2-6x...
please provide step by step solution
Thank you
(1 point) Use the inner product (5,8) = $* f()g(x) dx in the vector space P(R) of polynomials to find the orthogonal projection of f(x) = 2x2 + 4 onto the subspace V spanned by g(x) = x and h(x) = 1. (Caution: x and 1 do not form an orthogonal basis of V.) projy(f) =
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