Prove that if T is a unitary operator on a finite-dimensional inner product space, then T has a "square root"; that is, there exists a unitary operator U such that T = U².

Q: Prove that if V is a finite-dimensional vector space over a field F, then a subset {βi , β2 , . . .…

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Q: (d) If S,T are symmetric linear operators on an inner product space V, then their composition ST is…

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Q: There exists an ordered basis for V such that every operator

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Q: Consider the right-shift on the Hilbert space l2: S:l → l, S(@1,a2, ...) = (0, a1, a2, ...). Define…

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Q: Prove that if T is a unitary operator on a finite-dimensional inner product space V, then T has a…

A: Since, T is a unitary operator, there exists an orthonormal basis consisting of eigen vectors of T…

Q: 2. Let H H H where H be a Hilbert space. Let A = B(H) and B be the operator defined on H by 0 iA B =…

A: As per the question we are given a matrix operator B on the Hilbert space H. And we have to show :…

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A: Given:

Q: If H is a Hilbert space and A € BL (H) is positive, prove that A+21 is invertible.

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Q: Let T be a linear operator on a finite-dimensional vector space V with the distinct eigenvalues λ1,…

A: Let T be a linear operator on a finite-dimensional vector space V with the distinct eigenvalues λ1,…

Q: Let T be self-adjoint on a finite-dimensional inner product space V. Prove that Vr E V, ||T(x)± i||?…

A: The complete solutions are given below

Q: 14. Let T be a linear operator on a finite-dimensional inner product space V. Then T is self-adjoint…

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Q: Example (3): suppose X = F", with the inner product defined as (x, y) = E=1*iYt, Vx, y'E X, prove…

A: The objective is to prove that X is Hilbert space.

Q: ) Find the representation [T]B of T relative to the basis B'. Use [T]B to verify that T is…

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Q: Prove that k(x,x)=xAx' is a valid kernel, where A is a symmetric positive semidefinite matrix.

A: Let  we consider      kx,x' = xTAx'…

Q: (a) Let V be an inner product space with inner product (,), and let ~1,...Xn,J EV. Use proof by…

A: Note: Hi! Thank you for the question as per the honor code, we’ll answer the first question since…

Q: if T is an operator in a real vectorial space with characteristic polynomial (x²-2x+5)². What are…

A: For the solution follow the next steps.

Q: 7. Let T, and T, be bounded linear operators on a complex Hilbert space H into itself. If (T,x,…

A: Given: <T1x, x>=<T2x, x>, ∀ x ∈ H. To prove: T1 = T2.

Q: Let T be a self-adjoint operator on a Hilbert space H. Show that its norm is given by ||T|| = sup…

A: Given that T is a self-adjoint operator on a Hilbert space H. To prove: T=supx=1Tx,x.

Q: 20. Let V be a finite-dimensional inner product space and let U be a linear operator on V. Then U is…

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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: Determine whether the given map o is an isomorphism of the first binary structure with the second.…

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Q: Show that a bounded linear operator A on complex Hilbert space H is self adjoint if and only if is…

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Q: 2. Prove that every element in a vector space has a unique additive inverse.

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Q: Let A€BL(H) be a compact operator, then show that A*A is a compact

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Q: Let T be a linear operator on a vector space V, and let γ be a cycle of generalized eigenvectors…

A: Let T  be a linear operator on a vector space V.

Q: (a) Given a finite-dimensional inner product space V and a linear operator T:V V, explain how w…

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Q: Let V be a finite-dimensional inner product space over F. (a) Prove that the trace of every positive…

A: Below given clear explanation

Q: Assume thát B is a basis for an finite-dimensional inner product space V, and let a E V. Prove that…

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Q: If (e,} is an orthonormal basis for a Halbert space H, prove that || x ||2=£ | |² for all x in H.

A: An inner product space which is complete with respect to the norm coming out of the inner product is…

Q: (a) Given a finite-dimensional inner product space V and a linear operator T : V → V, explain how we…

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Q: Let V be a finite-dimensional inner product space, and let E be an idempotent linear operator on V.…

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Q: 3. Let A, B, T be bounded self-adjoint linear operators on a complex Hilbert space H. If T20 and…

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Q: Let B be an n ×n invertible matrix. Define Φ: Mn×n(F) →Mn×n(F) by Φ(A) = B−1AB. Prove that Φ is an…

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Q: Let H be a Hilbert space and A be a positive operator on H. Prove that I + A is invertible in BL…

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A: Given: L is the linear operator on and it is define by,

Q: Let V and W be finite-dimensional vector spaces, and let T: V → W be a linear transformation.…

A: Let T: V->W be a linear transformation. Β is a basis for V.It is required to prove that T is an…

Q: Let T be a linear operator on a finite-dimenstona! inner product space V. Let E1,..., Ex be linear…

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Q: Let V be a finite-dimensional inner product space and let U be a linear operator on V. Then U is…

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Q: Prove that the null space of an mxn matrix is a sub space of R^n

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Q: Let T be a normal operator on a finite-dimensional inner product space. Prove that if T is a…

A: By definition, given that T is a projection.

Q: Let X be a Hilbert space and T:X → X and T2: X → X are self-adjoint operators such that either…

A: Given we have given that X is a Hilbert space and T1 : X→X & T2 :X→X  are self -adjoint…

Q: Prove that if V is a two-dimensional real inner product space, then thecomposite of two reflections…

A: If two dimensional real inner product space, let T1 , T2 are a reflections on V then,…

Q: prove that A is invertible and A = B = C. %3D

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Q: Let U be a unitary matrix. Prove that (a) U is also a unitary matrix.

A: please comment if you need any clarification. If you find my answer useful please put thumbs up.…

Q: Let V be a finite dimensional inner product space and T a non negative operator on V. Then T has a…

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Q: Label the following statements as true or false. Assume that the underlying inner product spaces are…

A: (a) True.If T = T*, then T*T = T2 = TT*.

Q: Theorem 6.13. Let B be a basis for a space X. Then X is compact if and only if every cover of X by…

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Q: Let T be a self-adjoint operator on a finite-dimensional inner product space V. Prove that for all x…

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Q: Prove that a unitary transform leaves the inner product invariant. That is if U is unitary, then…

A: Please check step 2 for the solution.

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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