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

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

A:

Q: If the Wronskian of f and g is t cos t - sin t, and if u = f + 3g, v= fg, find the Wronskian of u…

A: It is provided that the Wronskian of f and g is tcost-sint. We need to determine the Wronskian of u,…

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

A:

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

A:

Q: Define an operator T: fºº → f° by T(5₁) = (¹) # and bounded. for each (₁) E. Show that T is linear…

A:

Q: please solve this linear algebra problem.

A:

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

A:

Q: Consider the linear operator defined by Show that T is compact. E1 E2 E3 T:l² l², x = (1, 2, 3, ...)…

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: Let D, be the linear transformation from C'[a, b] into C[a, b]. Find the preimage of the function.…

A:

Q: Let L:R R be a linear operator such that L X2 X3- X2 then Ker(L) spanned by: Select one: (e, e )

A:

Q: Let L:R → R3 be a linear operator such that L( X2 X1 then Im(L) spanned by: X3 X1 Select one: a. O…

A:

Q: .

A: Let L be the linear operator on R3 defined by

Q: d x"J,(x)=-x"Jns(x) d, n+1 X.

A:

Q: Let f(x, y, z) = x³e" Vz² + 5 and let x = rt², y = r2t², and z = sin(t) cos(r). (a) Draw a tree…

A:

Q: [2] Show that the functions f (x) = 4x" , and f,(x) = x*+* linearly independent (-00, 00). are on

A: To Show: Show that the functions f1x = 4xn, and f2x = xn+4 are linearly independent on -∞, ∞…

Q: Show f(x) = e2= and g(x) = x e²z linearly independent by finding its Wronskian.

A:

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: 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: A function e : X → Y is an embedding if one ond for V

A:

Q: Let LR R be a linear operator such that L X, )-| X1 then Ker(L) spanned by: Select one: a. (e2,e3)…

A:

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

A: The given operator A such that A:C0,1→C0,1. We use the inverse operator method for solving linear…

Q: each linear operator T on V, find the minimal polynomial of T. (d) V = Mnxn(R) and T(A)=At . Hint:…

A:

Q: Let T: X→→→→→ Y be a linear operator and dim X = dim Y=n<∞. Show that R(T) = Y if and only if T¹…

A:

Q: Let D, be the linear transformation from C'(a, b] into C[a, b]. Find the preimage of the function.…

A:

Q: 1. Determine whether the functions f1(x) = cos 3x, f2(x) = x, and f3(x) = cos² x are linearly…

A:

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

A:

Q: A E R"X", Q e R"Xn and Q > 0, C e R™×n. For a given X € R"Xn and X > 0, define the operator h as…

A: For a square matrix, the trace is the sum of the diagonal entries. Since diagonal can be obtained…

Q: Let L: R→ R be a linear operator such that L X2 )= X3-X2, then Ker(L) spanned by: X3 X3-X1. Select…

A:

Q: Let L: R→ R be a linear operator such that L X2 )= X3-X2, then L Select one: O a. b. O d.

A: Solution of question as follows

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…

A:

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: 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 L: R → R be a linear operator such that L X2 )=| X1 then Ker(L) spanned by: Select one: а. b.…

A: Given a linear operator L from R3 to R3. We need to find the set of vectors that spans Ker(L)

Q: Consider the following operator A : L²[0, 1] → L'[0, 1] A(f)(x) = (f(x))² + f(x). If this a linear…

A: Given, the operator  A : L20, 1→L10, 1 defind by                               A(f)(x)=(f(x))2+f(x)…

Q: {x} € 1, Show that property (3) of a matroid implies: For any S e there exists y € S, such that S \…

A: Following is the definition of a matroid. A finite matroid M is a pair (E,?), where E is a finite…

Q: Consider the following. w = In(x2 + y), x = 2t, y = 8 - t dw by using the appropriate Chain Rule. dt…

A: Solution is given below..

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

A:

Q: Let D, be the linear transformation from C'[a, b] into C[a, b]. Find the preimage of the function.…

A: Dx(f)=8x+5 Apply integration both side  ∫Dx(f)dX=∫(8x+5)dx   f(x)=(8x2)/2 + 5x +c (here c is the…

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 w f(t,0) where t and 0 are functions ofr and s, %3D dw is equal to Or then according to the…

A:

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…

A:

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