4. Show that, for an invertible linear operator T : V → V, (T-1)* = (T*)-1.

Q: The linearization at a = 0 to v8 + 5x is A+ Bx. Compute A and B. A = B = ||

A:

Q: Show that ex and e−x are linearly independent in C(−∞,∞).

A:

Q: Define: T:P, → R by T(P) = [ p(x)dx. What is the kernel of T ?

A: Given that: T:P2→RTp=∫01px dx

Q: Is {x² – 2.x, x²+x, x²} linearly independent dependent? Prove your answer. or

A: Answer

Q: Show that the operators T and T₂ from 2 R² into R² de fined by T₁: (F₁) √₂ ) → (9₂9₁) and T2=…

A:

Q: The linearization at a = 0 to /3 + 9x is A + Bx. Compute A and B. A = B =

A: Given: f(x)=3+9x

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

A:

Q: y/ + 2ry = 0, • State the order of the DE (1.1). • Classify the DE (1.1) in terms of linearity. •…

A: Since you have posted a question with multiple sub-parts, we will solve first three sub-parts for…

Q: The set of functions {Vx, x, x²} is linearly dependent on (0, oo). True False

A:

Q: find the linearization at x = a. f (x) = x4, a = 1

A: Given function: Using the formula of linearization of a function f(x) at x =a :

Q: 4. Show that for any bounded linear operator T on H, the operators T₁=1/√(T+T*) and T₁=1 (1 ==…

A:

Q: Show that if A is invertible and DA = CA, then D = C.

A: Given that, A is invertible and DA=CA.

Q: 2 Given t = -1 and U = -1|, for which is the set {u, T, w} {d, v, 2 linearly independent?

A: Given that, u→=1-12 and v→=2-11 We have to find the vector w→ for which the set u→, v→, w→ is…

Q: If f(x, y) is a potential function for the gradient vector field F(x, y) (Зх — 2у) і — (2л — у) ј, —…

A: Since F is conservativei.e, ∇×F  , so there exists some potential function f so that ∇f =F .

Q: The linearization at a = 0 to v4 + 4x is A + Bx. Compute A and B. A = B = %3D II

A:

Q: 25. Suppose that T: R2 → R² is a linear operator such that T. -1 | T. -1 Is it possible to determine…

A:

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: Let f(x, y) = 2x² + 5xy + 2y² + 5x – 5y + 2. Find the linearization L(x, y) of f(r, y) at the point…

A:

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

A:

Q: 8. Prove that the differential operator M: C² [a,b] → C[a,b] defined by d²u du Mu= 2+3u dx is a…

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

A:

Q: (b) Show that the Linear guen Operators. are not with T(f)= Lf6), f()+1].

A:

Q: 1. Use isoclines to sketch the direction field for y' = x+ 2y and sketch the solution curve that…

A:

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: Let L: R2 → R2² be the linear operator defined by L 2

A:

Q: 1 if i= j 0 otherwise. P:(c;) = 5 is a linearly independent set.

A:

Q: find the linearization at x = a. y = (1 + x2)^−1/2, a = 0

A: We find the derivative of the given function using power rule and chain rule

Q: 2. Let T: R³ → R³ be the operator given by -1 1 0 -1 0 0 Determine whether T is decomposable or…

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Q: s the linear operator T : R3 --> R3 defined by T (x, y, z) = (2x - 3y + 7z, 5y + z, -z)…

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

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Q: Calculate A and B so that t B %3D 1+1 (1-z²)

A:

Q: 10. Show that 1/x, x are linearly independent on (0, 0).

A:

Q: Consider a regularizing term defined by where k!2* and the linear differential operator D is defined…

A: We are given the following regularizing term.                    ∫ℝN0DF(x)2 dx=∑k=0∞ak∫ℝN0DkF(x)2 dx…

Q: 8. Show that an isometric linear operator T: H→→ H satisfies T*T= I, where I is the identity…

A: Let T : H → H  be an isometric linear operator. We need to show that , T satisfies T* T = I We know…

Q: suppose that x,y,z∈R3 are such that x.(y×z)≠0. prove that x,y,z are linearly independent.

A:

Q: 2. Let T : R³ → R³ be the operator given by -1 1 0 0 −1 1 0 0 – 1 - T(v) = = v. Determine whether T…

A: given matrix T=-1100-1100-1 determine whether it is decomposable or indecomposable.

Q: 1. Use the Wronskian to show that the functions are linearly independent. Yı = e*, y2 = e2*, y3 =…

A:

Q: find the linearization L(x) of ƒ(x) at x = a. ƒ(x) = x + 1/x, a = 1

A: Given fx=x+1x

Q: the function T:C+C defined by T(2) = z a linear transformation ?What if we consider C as a vector…

A:

Q: Conversely, if T :V V is a linear operator such that T2 = T, show that V = kerT O imT. [Hint: 3.b, v…

A:

Q: 19: Find two linear operator T, and T,on V(R) such that T,T, = 0 but T,7, - 0.

A:

Q: 4) Find Yp, using D-operator, where Yp (e). %3D (D+1) (D-2)

A:

Q: find the linearization at x = a. y = e√x , a = 1

A: Given function: Using the formula of linearization:

Q: 41. Let f(x, y, z) =x?y+ y²z. Use the Chain Rule to calculate af/as and af/at (in terms of s and f),…

A: Since f is function of x, y and z.  For partial derivative f with respect to s and t we using chain…

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

A:

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: Determine whether the following are linear operatorson Rn×n. L (A) = A − AT

A:

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