Let V = P, (R), equipped with the inner product defined by (f, 9) = | f(t)g(t) dt. Then {1.2/3(- с — x + is an orthonormal basis for V. (You do not need to verify this.) Let T:V → R be the linear functional on V defined by T(f) = f'(1/2) (i.e. T(f(x)) is the derivative of f(x), evaluated at x = 1/2). Find a polynomial h e V such that T(f) (f, h) for all f e V. Enter the coefficients of h(x) below. = h(x) = x2 + x + If any coefficient is not an integer, enter its decimal representation up to three digits after the decimal point.

Q: Consider the data: (zo, f(xo)), (₁, f(x1)), (2, f(x₂)), (#3, f(#3))).\\ The Lagrange basis…

A: Lix=x-x0x-x1·..x-xi-1x-xi+1·...x-xnxi-x0xi-x1·..xi-xi-1xi-xi+1·...xi-xn∴Lixj=0     for i≠j1     for…

Q: 20) For each vector field F, evaluate the circulation along the curve C consisting of the portion of…

A: Step 1:Step 2:Step 3:Step 4:

Q: 6. Let p(t) = ² + 1 and g(t) = t - 4 be vectors in P2 with inner product (f(t), g(t)) = f(t)g(t) dt.…

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Q: Q2) (a) Use the definition = ]f(t)g(t)dt in T0,1] to find the components of f(t)=2+t in the…

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Q: et L be the linear operator on R? defined by L(x1, x2) = (x1 + 2x2,3x1)* Find the matrix A…

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Q: 9. Let P3(C) be a vector space over C and let D: P3(C)→ P3(C) be the differential operator defined…

A: Since you have posted a multiple question according to guildlines I will solve first (Q9)question…

Q: 3. Let T : R² → R² be defined by T = X Y -x+y x-y

A: 3. Given that  be defined by:

Q: 1. Show that the given vector functions are linearly independent on (-∞, ∞). X₁ (t) = = 0 X₂ (t) H…

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Q: Let v=(1', 1, 3.), u=(0, -4', 1) are nonzero vectors in R³. Subtask (a). Whether a linear functional…

A: Functional Analysis:

Q: Create a vector of from F(x,y,z) such that the x, y, &z components contain at least two variables…

A: Given : vector of the form F(x , y ,z)  To Find : vector of the form F(x , y ,z)  such that the x, y…

Q: Let T: R> R a linear transformation defined by T(x, y, z) = (x+2y-z, y+2, x+y-2z). Find a basis for…

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Q: 12. Prove that if d divides a and d divides b, then d divides ax + by for all integers a and y.

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Q: Compute the result of counterclockwise rotation of a vector a = absin (a) 81 ∞as 23 by an angle = ³.

A: If a vector v is rotated counterclockwise about the origin through an angle θ then the resultant…

Q: V is the linear span of the polynomials a =x' + 2x – 2x + 1, ß=x' + 3x² - x+4, y = 2x +x - 7x – 7…

A:

Q: Let C1,2 be the vector space of continuous functions mapping the interval 1 <x<2 into R. Let T:…

A:

Q: Let A(t) = f(t)i + f₂(t)) and B(t) = g(t)i+g₂(t)), te[0, 1] where fi, f2, 81, 82 are continuous fun-…

A: Solution

Q: Determine if the scalar vector field is Continuous on the indicated domain f(x) = (x²+1) S=(0,0) Х…

A: f(x)=x2+1/x​ is not Lipschitz continuous on the indicated domain.Explanation:Approach to solving the…

Q: 2: Show that a map T is linear if and only if T(Au + v) = XT(u)+ T(v) for all A ER and vectors u, v.

A: A linear transformation satisfies  T:U-V  Then T(u+v) = T(u) +T(v)          T(au) = aT(u)  For all a…

Q: 8. Let P3(C) be a vector space over C and let D: P3(C) → P3(C) be the differential operator defined…

A: D: P3ℂ→P3ℂ be the differential operatorDpt=dpdtrelative to the basis 1, t, t2, t3

Q: show that the given vector functions are linearly independent on (−∞,∞).

A: To show that the given vector functions are linearly independent x1t=   et-et,        x2t=etet

Q: Find a singular value decomposition for the vector x = (1, 5, 7, 5)'.

A: Given: A vector Objective: To find the singular value decomposition (SVD) of the vector . The SVD…

Q: Consider the vector space Pg. We define linear functionals J«(p(z)) = p(-1), J2(p(x)) = p'(–1),…

A: The complete solutions are given below

Q: ¹([])=[+] 5. Let T: R2 R2 be a linear transformation defined by T a) Find a basis for Null(T) b)…

A:

Q: Find the domain of the vector function T (ty= Vt + 7 Domain: {t <t< }

A:

Q: 1. Let F : R? –→ R² be defined by F(x, y) = (4.x + 5y, 2x – y). (a) Find the matrix A of F relative…

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Q: Let T: R² R² be defined by T Let u = T(u): ¹([^])=[^²=²] = X2 H | amnet C = { v₁ = [!] > v₂ = [2]}…

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Q: Let Co,4, be the vector space of continuous functions mapping the interval 0 <x< 4 into R. Let T :…

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Q: Let P, denote the vector space of polynomials in the variable a of degree n or less with real…

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Q: Define T : R? → R by 7((x, y)) = x+y. Describe the kernel of and fibers of T geometrically.

A: The kernel is a line with slope -1 and passes through the origin.

Q: Consider the vector space of all real-valued differentiable functions, and let V denote its subspace…

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Q: Let P, denote the vector space of polynomials in the variable æ of degree n or less with real…

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Q: 33. Let V - P3 be the vector space of polynomials of degree <3, and let p₁(x) = (2-x)² and p₂(x) =…

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Q: Solve the given differential equation : dy x² – 2xy = 3y*, y(1)=1/2 dx

A: We will find out the required solution.

Q: Let P2(R) be the vector space of polynomials over R up to degree 2. Consider {1+x – 2x?, –1 +x+x², 1…

A: Consider the provided question, Given that,  B=1+x-2x2, -1+x+x2, 1-x+x2B'=1-3x+x2, 1-3x-2x2,…

Q: Let V be the vector space of all (2 x 2) matrices and define T: V → V by T(A) = AT. Show that I is…

A: