Let X = {−1,0,1} and consider the inner product on F(X) defined as follows: ⟨f, g⟩ = f (−1)g(−1) + 2f (0)g(0) + 4f (1)g(1). Let E be the subspace of all even functions defined on X. Set g(x) = x2 + x for every x ∈ X. Find the orthogonal projection of g onto E.

Let X = {−1,0,1} and consider the inner product on F(X) defined as follows: ⟨f, g⟩ = f (−1)g(−1) + 2f (0)g(0) + 4f (1)g(1). Let E be the subspace of all even functions defined on X. Set g(x) = x2 + x for every x ∈ X. Find the orthogonal projection of g onto E.

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter5: Inner Product Spaces

Section5.CR: Review Exercises

Problem 54CR: Let V be an two dimensional subspace of R4 spanned by (0,1,0,1) and (0,2,0,0). Write the vector...

See similar textbooks

Related questions

Q: 7. The following matrix is the augmented matrix for a linear system. 5 0 0 5 -1 0 1 0 3 1 2 0 0 1 4…

Q: (a) How many integers are there from 1,000 through 9,999? 9000 (b) How many odd integers are there…

A: We need to solve the given questions.

Q: Given det det det det a b C [iii] 7d 7 e 7 f hi 2 a b c C ghi de f a b c de f ghi D b D 38 CONCURSO…

Q: 1. Use the minimum-entry and u-v methods to solve the following balanced minimum-cost trans-…

A: SuppliesM4M3M2M122 6 6 10W₁5 4 3 30W2 4W3 16 9 6 20Demands 8102022Use Bland's rule to choose…

Q: Assume you are given recursive algorithms (like search algorithms) that, given large inputs of size…

Q: 3. Consider the function f(x, y, z) = sin(y) - √x² + 2². Find the gradient vector f(x, y, z) at Po =…

Q: #1. S is the part of the plane 5x - 4y + z = 2 that is above 0 ≤ x ≤ 1,0 ≤ y ≤ 1 in the x F = (x,…

A: Given that S is the part of the plane that is abovein the xy-plane.…

Q: Find the work done by the forcefield F(x, y) = (x + 2y³)j as an object moves once counterclockwise…

Q: Proof by contradiction ● Show for √5 Q.

Q: Find the dimensions of the following linear spaces. (a) The real linear space C7 (b) P₂ (c) The…

A: We have to find the dimensions of each of the following linear spaces.a) The real linear space b) c)…

Q: det b2 a1 92 261 262 b1 b3 az 2b3 + det C2 a1 92 2b1 262 C1 C3 az 263 = det b₁ b2 b3 a1 a2 a3 + det…

Q: (a) Suppose a country has a different target level of output than the ECB. Further assume that a…

A: In the context of the European Monetary Union (EMU), where a common currency and a single central…

Q: Find an orthogonal basis for the column space of the matrix to the right. An orthogonal basis for…

A: Here we have to find the orthogonal basis for given matrix.

Q: Definition of Infinite Limit: Let X CR, f: X → R and a E X'. If for every M > 0 there exists > 0…

A: Definition of infinite limit: Let X⊆R, f:X→R and a∈X′. If for every M>0 there exists δ>0 such…

Q: Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to…

Q: Problem 0.5 For each sequence [an}a1, find lim an by rational- izing the expression and determine…

A: (1) Given that…

Q: 16. If f is differentiable at xo, prove that lim h→0 f(xo + αh) - f(xo - ßh) h = (x + ß)ƒ'(xo).

Q: Proof: Let n be any integer such that n ≥ 2. [We will show that for every integer n ≥ 2, P(n + 1, 3)…

Q: Let A ≤R be Countable. Prove that RIA is countable.

A: Given thatA is a countable subset of a set of real numbers .We have to prove that \ A is countable.

Q: Question: The polynomials p(x) = x(x - 1) and q(x) = (x - 1)(x - 2) are orthogonal when the inner…

Q: Use Newton's method to approximate a root of the equation 5x³ + 7x + 2 = 0 as follows. Let x₁ = -1…

Q: Part I: Consider the subspace of M2x2 that has [23] 32 is the coordinate vector of Part II: What…

Q: Laplace Transform of f(t) = eªt sin(wt) о O (s-a)²+w² ພ (s-a)²+w² S (s-a)²+w² W (s-a)+w s-a (s-a)+w

A: Given function isWe have to find the Laplace transform of given function .We know thati) If , then…

Q: prove that 11 ㅣㅣㅣㅣ 리 ZtZ3 < 1211 12월 18

A: For complex numbers z2,z3 with

Q: Evaluate f 3xdA where R is the region shown. 3. 2 4 2 10 R 2 4. 0 R

A: We have to find the double integral of problem (4).

Q: Solve the Linear 1st Order O.D.Es:

A: Given that the differential equation is

Q: SOLVE THE FOLLOWING PROBLEM AND SHOW YOUR COMPLETE AND DETAILED SOLUTIONS FOR BETTER UNDERSTANDING.…

Q: Find the characteristic equation of the matrix 9 -10 6

Q: f is a 27-periodic function with the following Fourier coefficients ao = 0 a₁ = b₁ = 0 a2 = 0 b₂ = 0…

A: Let given that the coefficient of Fourier seriesa0=0a1= 1/3b1=0a2 = b2 =0 = a3 = b3Then series…

Q: Question 1: Consider the subspace of M2×2 that has { 1 [01] $11 with respect to this basis? What is…

Q: Find the Laplace transform of e-at u(t) O 1/(s-a) O 1/s O 1/(sa) O 1/(s+a)

Q: SOLVE THE FOLLOWING PROBLEM AND SHOW YOUR COMPLETE AND DETAILED SOLUTIONS FOR BETTER UNDERSTANDING.…

Q: 1. Complete the following table Relation y=(x+5)²-6 {(-5,4),(-5,3),(2,5)} Function (yes/ no) Domain…

Q: 3. (Section 17.6) Determine the upward flux of F = through the part of z = x2 + y2 below z = 2.

Q: This information can be encoded into a 2 x 2 matrix, as follows: [0.7 = [0.3 0.2 0.3 0.8 P = (a)…

A: Compute the vector Given that,general formula to find the inverse of 2 by 2 matrix,Now apply this…

Q: 14. Find the general formula for the following sequence 4, 12,36... 15. You take a loan on a $12000…

A: We have to solve the question 14 and 15 The solution of both the questions is given below

Q: Prove: If α∊R, then the function αf is differentiable at c, and (αf)'(c)=αf'(c)

Q: Consider the basic feasible solution of a transportation problem given below: 14 12 10 13 11 U1 15…

Q: Construct the Taylor series for f at x0=0.

A: We need to construct the Taylor series for f at x0=0.

Q: 1. Find the maximum and minimum values of f (x, y) = x² + 3y + 7 on the set R, where R is the region…

A: We have to find the maximum and minimum values of f(x, y) = x2 + 3y + 7 on the set R, where R is…

Q: Assessed Question 4. Suppose you roll a pair of dice until you get a double-six. What is the…

A: According to the given data, find the smallest number of times must be roll the pair of dice at…

Q: 3 Given the function f(x) = x + 3 on the interval 0 ≤ x ≤ 1 the Fourier sine series expansion of…

Q: (12) For the polynomial p(x), if p(3) = 0, it can be concluded that A) x+3 is a factor of p(x) x-3…

Q: Problem 0.4 diverge: Determine whether the following series converge or 3 n=0 100+n (1) ΣΩ (2) -99…

A: Since you have posted a question with multiple sub-parts. We will provide the solution only to the…

Q: SOLVE THE FOLLOWING PROBLEM AND SHOW YOUR COMPLETE SOLUTIONS FOR BETTER UNDERSTANDING. f(x) 0 0 100…

Q: Can you help me with this? Thanks.

A: The objective of the question is to determine if the statement 'If x squared plus 5x is less than 0…

Q: Use the binomial theorem to expand (2c − 3d)^4 discrete structures (counting and probability unit)

Q: Suppose that a linear program with bounded feasible region has I optimal extreme points v1, ..., vl.…

Q: Question 1 The process for handling insurance claims at a branch of the Green Insurance Company has…

Q: Is it necessarily the case that for a graph with n vertices and exactly (2¹) +1 edges be…

A: The statement you provided is not necessarily true.If n vertices and exact edges does not guarantee…

Question

Let X = {−1,0,1} and consider the inner product on F(X) defined as follows:

⟨f, g⟩ = f (−1)g(−1) + 2f (0)g(0) + 4f (1)g(1).

Let E be the subspace of all even functions defined on X. Set g(x) = x² + x for every x ∈ X.

Find the orthogonal projection of g onto E.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1: Now.

VIEW

Step 2: Then

VIEW

Solution

VIEW

Step by step

Solved in 3 steps with 3 images

SEE SOLUTION Check out a sample Q&A here

Similar questions

Let f1(x)=3x and f2(x)=|x|. Graph both functions on the interval 2x2. Show that these functions are linearly dependent in the vector space C[0,1], but linearly independent in C[1,1].
Find an orthonormal basis for the subspace of Euclidean 3 space below. W={(x1,x2,x3):x1+x2+x3=0}
Let T be a linear transformation from R2 into R2 such that T(1,0)=(1,1) and T(0,1)=(1,1). Find T(1,4) and T(2,1).
In Exercises 1-12, determine whether T is a linear transformation. T:FF defined by T(f)=f(x2)
Let T be a linear transformation T such that T(v)=kv for v in Rn. Find the standard matrix for T.
Let T:R3R3 be the linear transformation that projects u onto v=(2,1,1). (a) Find the rank and nullity of T. (b) Find a basis for the kernel of T.
In Exercises 24-45, use Theorem 6.2 to determine whether W is a subspace of V. V=F,W=finF:f(x)=f(x)
In Exercises 24-45, use Theorem 6.2 to determine whether W is a subspace of V. 34. ,
Let V be an two dimensional subspace of R4 spanned by (0,1,0,1) and (0,2,0,0). Write the vector u=(1,1,1,1) in the form u=v+w, where v is in V and w is orthogonal to every vector in V.
Find a basis B for R3 such that the matrix for the linear transformation T:R3R3, T(x,y,z)=(2x2z,2y2z,3x3z), relative to B is diagonal.
Let T be a linear transformation from R3 into R such that T(1,1,1)=1, T(1,1,0)=2 and T(1,0,0)=3. Find T(0,1,1)