9. Show that a bounded linear operator T: H–→H on a Hilbert space H has a finite dimensional range if and only if T can be represented in the form Tx = 2 (x, v;)w; [v, w, e H]. j=1

Q: 3. The linear operator T: R³ R³ is defined by T(x1, x2, x3) = (W1₁, W2, W3), where w₁ = 2x₁ + 4x2 +…

A:

Q: Given points pı = , and p3 in R², P2 let S = conv{pj,P2»P3}. For each linear functional f, find the…

A:

Q: Define a function f : C -> C by f(x+iy) = (x+2y) + i(3x+4y) for x,y in R. Show that f is additive…

A:

Q: Let T : V → V be a linear operator over a finite dimensional vector space V . Prove that dim(ker(T))…

A:

Q: In the space C[0, π] let f, g, h, and j be the vectors defined by f(x) = 1 g(x) = x h(x) = COS X…

A: In the space C[0, ],

Q: Given points p = and p3 = in R?, 2 P2 = let S = conv {p],P2» P3}. For each linear functional f, find…

A:

Q: Let H1 and H2 be any two Hilbert Spaces and T : H1 → H2 be a bounded linear operator. Let M1 = N(T)…

A: Given:- Let H1 and H2 be any two Hilbert Spaces and T : H1 → H2 be a boundedlinear operator. Let M1…

Q: Question 4 inner product on V. Show that the following are equivalent: (1) ||v – w|| = ||S(v) –…

A:

Q: 3. Let V denote the vector space of all functions f: R" → R, equipped with addition + : V × V → V…

A:

Q: . Let B(X, Y) denote the space of all bounded linear operators between the normed spaces X and Y.…

A:

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: 3. Let V denote the vector space of all functions f: R" → R, equipped with addition + : V x V → V…

A:

Q: Verify that the set C[a, b] of all continuous real-valued functions defined on the interval a ≤ x ≤…

A:

Q: Let V be a vector space and suppose there exist maps S, T = L(V, V) such that range(S) ℃ null(T).…

A: Let V be a vector space and suppose there exist maps S,T:V→V such that .We need to prove that the…

Q: 3c

A: Step 1:a) To evaluate the adjoint (or conjugate transpose) of the linear operator T on the inner…

Q: 8. Suppose T: V V is a linear operator on an inner product space V. If T has an adjoint (hence, a…

A: Given that T is a linear operator on an inner product space V. If T has an adjoint. The unique…

Q: (a) ]Find all vectors in the form x= X2 x3 such that T(x) is a scalar multiple of x.

A: I am solving the first part for you.The given linear transformation is : from To find all vectors…

Q: 7. Let V be the vector space of polynomials in two variables x and y of degree at most two: V = {ax²…

A: 7. Let  be the vector space of polynomials in two polynomials x and y of degree at most two:.Let T…