Let T be the linear functional defined on C [a b] (with sup norm) by T (x) = ex (t) dt for x eC[a, b] Find || T ||

Q: Calculate the norm of the linear functional 1 į t f(t) dt on each of the spaces 0 (C[0, 1], || ||)…

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Q: Let ⁄2 be a bounded interval. For any g € L²(2), define the operator T, by T,(f) = f Then T, is a…

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Q: Let P(z) = | tin(t) dt. Use part 1 of the fundamental theorem of calculus to determine if P(z) is…

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Q: (b) Let f(t) = (t- 2)*(t +1)² be the characteristic polynomial for a linear operator T. Obtain all…

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Q: Find a norm of the function f(t) = t² in space C" 0, 1, where norm is given by formula max|f(t)|+…

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Q: Let f(t) = (t – 2)*(t+1)² be the characteristic polynomial for a linear operator T. Obtain all…

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Q: 3. Let L be a linear operator on R and let L(1) = a . Show that L(x) = ax for all x E R .

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Q: The mapping $ (y) = fy' : H→H* defined by is not a linear m

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Q: 3. (a) Show all functions satisfying u"(x) = 0 are linear by integrating the expression of u"(x) %3D…

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Q: Let W = F(x , y , z) be a function such that Fx(0 , 0 , 0) = -1 , Fy(0 , 0 , O) = 3 , F2(0 ,0,0) = -…

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Q: Find the linearization L (x, y) of the function f (x, y) = y/-x at (-9,4). 2 L(x, y) -x+ 3y – 6 3…

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Q: 3. Show that every y E lị defines a continuous linear functional on l through the inner product,…

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Q: Prove that if r is a vector-valued function that is continuous at c, then || r || is continuous at…

A: Consider the vector r→a=p(a)i+q(a)j+s(a)k.   If r is continuous at a=c. So, lima→cr→(t)=r→(c) Then…

Q: The function f (x, y) = x3y + x2 + y2 − 2x − y + 4 has a stationary point (1, 0). Compute the…

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Q: Let A(x) = S f(t) dt, with f(x) as in figure. 1. 1. 0,5 -1 -2 A(x) has a local minimum on (0, 6) at…

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Q: Let A = → Rm be (a;j)ij be an m xn matrix of real numbers, and let f : R" defined by x2 X2 f = A…

A: I am solving this in step 2

Q: [Exercise T1.13.] Prove that for any map f : R → R, a source has sensitive dependence on initial…

A: To Prove that for map f:ℝ→ℝ, a source has sensitive dependence on initial conditions:Firstly,Let x0…

Q: Define f : X → K by f(x) = x'(1), x € X. Again, the linear functional f is not continuous. By 6.4,…

A: Given: f:X→K by fx=x'1, x∈X To prove f is not continuous using the given corollary.

Q: Looking Ahead Use the Wronskian to show that {e' , te' I is linearly independent on R.

A: Consider the equation

Q: If S and T are normal linear operators satisfying ST* = T* S and TS* = S*T, show that their sum S+T…

A: I have used just definition of normal operator

Q: (c) Define f on C[0,1} by : f (x) = ] x(t)dt, xeC([0,1]. Prove that f is a bounded linear functional…

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Q: Let f(x, y) xy, x = 1- Vt, and y = 1+ Vt. Find df using the chain rule. dt

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Q: Determine the Wronskian of the given set of functions and determine whether they are linearly…

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Q: Determine if the functions Y1=e^{4x}senx and Y2=e^{4x}cosx are linearly independent or linearly…

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Q: fi (x) = x, f2(x) = cos x, fa(x) = 5

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Q: a) Define a linear operator on R' by performing the following operations in order: i) a shear in the…

A: We are authorized to answer three subparts at a time, since you have not mentioned which part you…

Q: 3 Let A be the following linear functional on C([0, 1]): xf (Va) da, f e C([0, 1]). Prove that A is…

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Q: Let A: C [0,1]→C [0,1]. Prove that operator A is continuous reversible, find A!, 2.11. Ax(1) = x(1)…

A: We have given operator A such that A: C[0,1]-C[0,1].Also we must know that the inverse operator…

Q: If T: H→→H is a bounded self-adjoint linear operator and T‡0, then T"#0. Prove this (a) for n = 2,…

A: Given T:H→H is a bounded self-adjoint linear operator and T≠0. We know that if T is a self adjoint…

Q: Consider the bounded linear operator T:C0,1]C0, 1] defined by -Ta =y y(t) = ta(t). Enter the norm of…

A: Given that T:C0,1→C0,1 is a bounded linear operator. Defined by x↦Tx where x is continuous function…

Q: Determine the extreme of the following functional: T/2 J(x) = | [x{(t) + * (t) + 2x1 (t)x2 (t)] dt…

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Q: Exercise 11.3 Prove the equality [* | f(x) dx = fm({x : [ f(x)| ≥ t}) dt, 0 where m is Lebesgue…

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Q: Let Te B(H) be an invertible operator. Show that a(T-) = 0(T)-.

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Q: Let A: C [0,1]→C [0,1]. Prove that operator A is continuous reversible, find A. Ax() = x() + ja-m}xt…

A: Solution:- We have given operator A such that A: C[0,1]- C[0, 1] Also we must know that the inverse…

Q: Let T be function from R to R2 de fined as T (x) = (0, x), Then show that if T is linear operator or…

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Q: Given a function f(x) = f(x1, x2) = (x2 − x 2 1 )(x2 − 2x 2 1 ), where x ∈ S =  [x1, x2] T : x1 ∈…

A: Given a function fx=fx1, x2=x2-x12x2-2x13, where x∈S=x1, x2T: x1∈ℝ, x2<0. (a) Find out the nature…

Q: Let f1(x) = 3x and f2(x) = ∣x∣. Graph both functions on the interval −2 ≤ x ≤ 2. Show that these…

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Q: (b) {T } is a sequence of bounded linear maps on a Banach space X and Tx = lim 1x exists for all xE…

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Q: 1 Are the following sets of functions, each defined on x E R, linearly independent? Use Wronskian.…

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Q: Let V=P(R) and for j≥1 define Tj(f(x))=f(j)(x), where f(j)(x) is the jth derivative of f(x). Prove…

A: let n be any integer and a1,a2,...an be scalars such that  a1T1+a2T2+a3T3+...+anTn=0 where T is the…

Q: Let T: C* (R) → C (R), T(f) = f' – 2f be a (linear) differential operator on the space of infinitely…

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Q: Let T be function from R to R² de fined as T (x) = (x, 0), Then show that if T is linear operator or…

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Q: Let f(x, y, z) = x2y + y2z. Use the Chain Rule to calculate af /as and af /at (in terms of sand t),…

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Q: Find the Wronskians of the given sets of functions and determine whether the functions are linearly…

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Q: Give an example of a linear operator T on an inner product space V such that N(T) ≠ N(T∗).

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Q: Use the Wronskian to show that the given functions are linearly independent on the given interval I.…

A: For linearly independent, we show Wronskian is non -zero.

Q: Show that each linear operator T on R2 is nonsingular and find a formula for T- , where (a) T(x,y) =…

A: We have to show that linear operator T on R2 is nonsingular andfind the formula for…

Q: Let c(t) Use the chain rule to compute (foc)'(t). (cos(t), sin(t), t²) and f(x, y, z) = x² + y² +…

A: The objective is to find the derivative.

Q: (b) If f is a bounded linear functional on a Hilbert space H, show that there is a unique y e H such…

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Q: For a linear operator T on an inner product space V, prove that T∗T = T0 implies T = T0. Is the same…

A: Let T be a linear operator defined on an inner product space V.To prove