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Homework Answers
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(5a) Compute ([1,2], [0, 3]) for the inner product (x,y) =...
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...
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.
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...
Question 3. Let 3 5/' and for x(2),y -(,) ER2 define (a) Show that the assignment (x, y) > (x,y) defined ın (1) us an nner product [10 marks (b) If a - (1,-1) and b - (1,1), then show that the...
Question 3. Let 3 5/' and for x(2),y -(,) ER2 define (a) Show that the assignment (x, y) > (x,y) defined ın (1) us an nner product [10 marks (b) If a - (1,-1) and b - (1,1), then show that the vectors a and b are lınearly ndependent but they are not orthogonal with respect to the inner product n (1) 3 marks] (c) Given the vectors a and b in (b), the set (a, by is hence a...
interval-1,1. If f.geCL1.], we'l 7) The field of play is Cil the space of all functions that are continuous on...
interval-1,1. If f.geCL1.], we'l 7) The field of play is Cil the space of all functions that are continuous on the define the inner product as (f,g)= f'f(x)g(x)dx. The question is simply this: Find the orthogonal projection of e" onto P, and graph both functions on [-2,2].
interval-1,1. If f.geCL1.], we'l 7) The field of play is Cil the space of all functions that are continuous on the define the inner product as (f,g)= f'f(x)g(x)dx. The question is simply this:...
5) In C.), with inner product <f,g> [f(x)g(x)dx, let f(x) = x², g(x)= x', a) Compute<...
5) In C.), with inner product <f,g> [f(x)g(x)dx, let f(x) = x², g(x)= x', a) Compute< x², x? >; 0 b) Find the “angle” between the two functions.
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.
3. Let R be equipped with the inner product (x,y) = AX Ay, where A is...
3. Let R be equipped with the inner product (x,y) = AX Ay, where A is the matrix shown below: TO-4 21 A = 3 2 LO 0 5) a.) (5 points) Let v = (1,-1,3). Find || V || 1 b.) (5 points) Let x = (2,3,0) and y = (-3,2,1). Are x and y orthogonal in this inner product space? Justify your answer
this is all information was given, what do you need more? 2. Consider the inner product...
this is all information was given, what do you need more?
2. Consider the inner product space V = P,(R) with (5.91 = 5(0)g(t) dt, and let T: VV be the linear operator defined by T(f) = xf'(x) +2f(x). (1) Compute T*(1+2+x2). (ii) Determine whether or not there is an orthonormal basis of eigenvectors 8 for which [T], is diagonal. If such a basis exists, find one.
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...
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.
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...
Question 3. Let 3 5/' and for x(2),y -(,) ER2 define (a) Show that the assignment (x, y) > (x,y) defined ın (1) us an nner product [10 marks (b) If a - (1,-1) and b - (1,1), then show that the vectors a and b are lınearly ndependent but they are not orthogonal with respect to the inner product n (1) 3 marks] (c) Given the vectors a and b in (b), the set (a, by is hence a...
interval-1,1. If f.geCL1.], we'l 7) The field of play is Cil the space of all functions that are continuous on the define the inner product as (f,g)= f'f(x)g(x)dx. The question is simply this: Find the orthogonal projection of e" onto P, and graph both functions on [-2,2].
interval-1,1. If f.geCL1.], we'l 7) The field of play is Cil the space of all functions that are continuous on the define the inner product as (f,g)= f'f(x)g(x)dx. The question is simply this:...
5) In C.), with inner product <f,g> [f(x)g(x)dx, let f(x) = x², g(x)= x', a) Compute< x², x? >; 0 b) Find the “angle” between the two functions.
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.
3. Let R be equipped with the inner product (x,y) = AX Ay, where A is the matrix shown below: TO-4 21 A = 3 2 LO 0 5) a.) (5 points) Let v = (1,-1,3). Find || V || 1 b.) (5 points) Let x = (2,3,0) and y = (-3,2,1). Are x and y orthogonal in this inner product space? Justify your answer
this is all information was given, what do you need more?
2. Consider the inner product space V = P,(R) with (5.91 = 5(0)g(t) dt, and let T: VV be the linear operator defined by T(f) = xf'(x) +2f(x). (1) Compute T*(1+2+x2). (ii) Determine whether or not there is an orthonormal basis of eigenvectors 8 for which [T], is diagonal. If such a basis exists, find one.
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