Let X=L² [0, 1] and peLº [0, 1] and A be the operator on X given by Ax =4x, xeX show that the operator B : X→X defined by Bx = 6x, xeX, is the adjoint of A. Where 5 is the complex conjugate of o.

Q: 1. Let f(r1, 22, r3) = x1 sin(r23). Compute the following: %3D a) Dif(x1,x2, x3) b) D3f(x1, x2, x3)…

A:

Q: 1. Let f(x+iy) = u(x, y) + iv(x, y), where u : R² → R and v : R² → R are defined by u(x, y) = x +…

A:

Q: Let W = {(x, y) E R² : x, y > 0}. Define addition + of any two elements (x, y), (w, z) E W by (х, у)…

A:

Q: 7. In A ABC,X is the centroid. A A. If CW = 15, find CX and XW. W B B. If BX = 8, find BY and XY. C.…

A:

Q: Let xe c and ye show that <xiy= y AX is qu inmer product if A is hermition positive del iniite

A: (Solving the only one question as per our guidelines) Question: Let x∈ℂm and y∈ℂm Show that x,y=y*Ax…

Q: Give V = C([0, 1]) the inner product defined as follows: (f, g) :| f(t)g(t) dt for all f, g e V.…

A:

Q: Let LR→ R be a linear operator such that L X, )=| x, then Im(L) spanned by: Select one: a. (0,e,,e2)…

A:

Q: Determine if the ODE is a homogeneous DE. If it is, state its degree. 1. у' 2ху2 yx2 +y3 -4/(x³y3)

A:

Q: Suppose f : Z → Z is defined by f(x) = 5æ – 2. O a. fis invertible and its inverse is f-' = 5x + 2.…

A: To be invertible a function have to be bijective i.e. one to one as well as onto

Q: a) Let f: Z Z be such that f(x) 2*x. Is f invertible, and if so, what is its inverse?

A: # we are entitled to solve one question at a time, please resubmit the other question if you wish to…

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

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

Q: (a) Define the symbols ôf/DE and af/dz by af 1 (af 10f 2 (dr af 1 (of 18f i by, dz 2 dr' i dy as…

A: Since you have asked multiple questions, we will solve the Question (a) for you. If you want any…

Q: Let Ps have the inner product given by evaluation at-2,-1, 0,1, and 2. Let p(t) = t2- 2t and q(t) =…

A: Let P3 have the inner product given by evaluation at -2, -1, 0, 1 and 2 . Let p(t) = t2-2t  and q(t)…

Q: Consider the D-operator P(D) = mD² + 2kD +¢, where m and c are positive. If -c/m < k< 0, then

A:

Q: Let (An) and (Bn) be (Fn)-martingales on (2, F, P) and let Cn = A - Bn. Show that E [(Am - An) |Fn]…

A: Solution:

Q: Let x, y € R" be given. (a) If ||z||||y||=1 and z-y = 1, show that z = y. (b) If ||z|| ≤ 1 and ||y||…

A:

Q: 8. Prove the following properties of the Fourier convolution: (a) f (x) * g (x) = g (x) * f (x), (b)…

A: As per our company guidelines we are supposed to answer only first 3 subparts, kindly repost other…

Q: ; Let x, y ER" be given. (a) lf ||x|| = ||y|| = 1 and x - y = 1, show that x = y. (b) If ||x|| ≤ 1…

A:

Q: 39 In Z,[x], let I = (r + x + 2). Find the multiplicative inverse of 2x + 3+I inZ.[x\/I.,

A: Given:         In Z5x , I =x2+x+2   To find multiplicative inverse of 2x+3+I is   By dividing…

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: 3. Let K(r, y) = xy(x+ y), 0 < x,y <1 and consider the following operator K K f(y) = | K(y,…

A: As per the question we are given a kernel function K(x,y) = xy(x+y) for 0≤x,y≤1 And a operator k.…

Q: The integral (r+2y)e-dedy can be written and evaluated after using the transformation u = r+2y and v…

A:

Q: Evaluate z'dz where C is z=2 by Residue theorem. (z-1) (z-3) (s-2) ,(1-2)}

A: Given A complex integral we can use cauchy's residue theorem to solve the problem

Q: Let f : (0, 1] → (0,1) be the function defined by f ) = „ for all n E N, and f(x) = x if æ E (0, 1]…

A:

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: THEOREM 23. Let K be an integral operator on L2([a, b], a) KS = [°xe K(x, y)f(y) da(y) where T| K(x,…

A: We will use the basic knowledge of functional analysis to answer this question. Recall the…

Q: 2n Jnsi (x) = " J,(x) – J(4)

A:

Q: Let (S,F,u) be a mee e B₁, B2 E F satisfying (E 1) = μ(B₂) for any pair of B₁) = 0. Then ao-algebra…

A:

Q: Verify that the functions f1(x) = 1, f2(x) = sin x, and f3(x) = cos x are orthogonal in [−π, π], and…

A:

Q: Let T: X X be the linear operator defined as K = 4. if N is even, K = , if N is odd, %3D Tx(t) = a2)…

A:

Q: 2. Consider an operator of the form (Lf) (x) = Ef(x;)g;(x) where a O for all a E [a,b] and for all…

A:

Q: ) Let A. Cse le given by A(x, ウ2tm.) = (M年 Isthe operator e invertible ? If the anserer is He…

A:

Q: Let a ∈ R[[x]] with a0 not equal to 0. Prove that a has a multiplicative inverse in R[[x]]. You may…

A: We have given that , a∈ℝx a = a0 + a1x + a2x2 + ......  where each ai ∈ ℝ. Since , 1 ∈ ℝx  is an…

Q: Q6. Let C[0,1] have the inner product < f,g 2 S,"f(x)g(x)dx and let fn= cos nx (n=0,1,2,3,....).…

A:

Q: Let f(x) = sin x and g(x) = cos x, and W = span{f,g}. Use the Wronskian to show that f and g are…

A:

Q: Let F(x, y, z) - z* + x+y sin x + x In z- 0, where z- f(x, y), then at (0,0,1) and are az %3D %3D…

A:

Q: Let g C[0, 1] be a fixed function. Consider the operator A = B(C[0, 1]) defined by (Au) (s) :=…

A: Compact Operator: A compact operator is a linear operator A:X→Y where X, Y are normed vector spaces…

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

A:

Q: -i log(U) is Hermitian and therefore that 4. For a unitary operator U, show that T = U = el for some…

A: Let U be a unitary operator.  To prove that T=-ilogU is Hermitian.  Since, U is an unitary operator,…

Q: Let E/F be Galois and f e F[r] irreducible. Prove that any two irre- ducible factors of ƒ € E[x]…

A: Given : E | F is a Galois extension and f ∈ F [x]  is irreducible over F .  To prove : Any two…

Q: [0, 1]. If f(1), g () e V then show that (f(t), g (t)) = [ƒ(t) g (f) dt defines an inner product on…

A:

Q: Consider the operator T: 1 given by T(r1, 22,. ) = (0, r1, 22,. ). Is T a compact operator? Explain…

A:

Q: Let n E Z. Show that f(x) = x3 – nx +2 is irreducible over Q for %3D n + -1,3, 5.

A: We shall find all n ∈ Z such that f(x) is reducible. Assume that f(x) is reducible. As f(x) is…

Q: Let z = g(x, y) = x² + xy³ -y - 6x - 2y +4, where x = r cos (8) and y = r sin(8). дz Use the chain…

A:

Q: Let L:R→ R be a linear operator such that L X2 )=x1, then Im(L) spanned by: -> X3 Select one: a. 一 O…

A: The given linear operator is: L:ℝ3→ℝ3 such that Lx1x2x3=x1x1x1

Q: Show that The Hermite polynomials can be difined by the integral representation H, (x) = (-i)"2"e**…

A:

Q: If Ø : R3 → R3 be defined as Ø (x,y,z) = (2x, 4x –y,2x+ 3yz) then show that Ø is invertible. %3D

A: Given that ϕ:R3→R3 is defined as ϕx,y,z=2x,4x-y,2x+3yz. Now, ϕ is invertible if ϕ is one-one and…

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

A:

Q: determine where the given complex mapping is conformal. 3. f(z) = z + e + 1 5. f(z) = (z – 1)/2 4.…

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