Let T be a linear operator on an inner product space V. If (x), y>= 0 for all x, y ∈V, prove that T = T0. In fact, prove this result if the equality holds for all x and y in some basis for V.

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: Let X be a normed linear space and T: X→ X and S: X → X be two bounded linear operators, then SoT…

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Q: 15. If T is a self-adjoint linear operator on a finite-dimensional inner product space V, then detT…

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Q: Let V be an inner product space, and suppose that T: V→V is linear and that ||T(x)|| = ||x|| for all…

A: Given:

Q: Give an example of a linear operator T on R2 and an ordered basis for R2 that provides a…

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Q: Let T be a linear operator on a finite dimensional vector space, V and suppose that the different…

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Q: Let T be a linear operator on a finite-dimensional vector space V. Prove that the following…

A: An invariant subspace W of T has the property, such that vectors v belongs to W.

Q: Let T be a linear operator on R2 defined as follows on the standard basis of R2. T(1, 0) = (3, 2),…

A: From the given information. T is a linear operator on R2. The basic vectors are as follows.…

Q: Let T be a linear operator on an inner product space V, and supposethat IIT(x)ll = llxll for all x .…

A: Let us consider T be a linear operator on an inner product space V, and suppose that IIT(x)ll =…

Q: Let ẞ= {v1, ••• , vn} be an orthonormal basis for Rn. (a) Prove that, for any x and y in Rn, x · y =…

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Q: Let V be a finite-dimensional vector space over C, T in L(V) be a linear operator on V, and p(z) in…

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Q: Let B be an invertible matrix in M (R). Define T: M (R) → M_ (R) by nxn T(A) = B AB. Prove that T is…

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Q: Let H be a Hilbert space, then a closed linear subspace Mof H reduces an operator T if and only if M…

A: To prove this theorm firstly we prove the theorm which states .. A closed linear subspace M of a…

Q: Prove that the restriction of a linear operator T to a T-invariant subspace is a linear operator on…

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

A: It is given that T is a unitary operator on a finite dimensional inner product space.

Q: Define the following subset of R3 x1 E R³ : a'x= 0 = < x = x3 where a' = [-1,1,4]. Show that W is a…

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

A: Since we only answer up to 3 sub-parts, we’ll answer the first 3. Please resubmit the question and…

Q: Let T be a linear operator on a two-dimensional vector space V and suppose that T≠c| for any scalar…

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

Q: Let V be an inner product space, and let T and U be linear operators on V. Then (a) (T + U)∗= T∗+ U∗…

A: Since we only answer up to 3 sub-parts, we’ll answer the first 3. Please resubmit the question and…

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 β be a basis for a finite-dimensional inner product space. (a) Prove that if &lt;x, z&gt;= 0 for…

A: Let β be a basis for a finite-dimensional inner product space.

Q: Describe all linear operators T on R2 such that T is diagonalizable and T3 - 272 +T= To.

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Q: Let V be a finite dimensional vector space, with S : V → V a linear operator, and ⟨·, ·⟩ an inner…

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Q: Let T be a diagonalizable linear operator on   V -finite-dimensional, and let m be any positive…

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Q: • Exercise 3 Prove that if S and T are both bounded linear operators from the normed space X to the…

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Q: Let T be a linear operator on a vector space V, and let W1,W2, . . . ,Wk be T-invariant subspaces of…

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Q: LetT be a linear operator on an inner product space V. Let U1 = T + T*and U2 = TT*. Prove that U1 =…

A: T is an operator in an inner product space V over the field F together with an inner product. V×V→F…

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

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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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Q: Let T be a linear operator on a finite-dimensional vector space V such that the characteristic…

A: Let T be a linear operator on a finite-dimensional vector space V such that the characteristic…

Q: .

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

Q: im(T1 + 14T) im(14T† + T2).

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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 T: H- » H be a bounded positive self-adjoint linear operator on a complex Hilbert space. Using…

A: As per the question we have to show that the Cauchy-Schwarz inequality holds for the Hilbert space H…

Q: Let V be an inner product space, and let y, z ∈V. Define T: V →V by T(x) = &lt;x, y&gt;z for all x…

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Q: Let T be a linear operator on a finite-dimensional Prove that T is normal if and only if omplex…

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Q: Let T be a linear operator on a finite dimensional vector space V, and Let W1,W2,.....Wk be T…

A: Assume that T is diagonalizable. Then there exists an ordered basis of V, v1,v2,...vk  such that…

Q: Let A be a bounded linear operator on a Hilbert space H and ||A(x)||=||x|| ¥xe H. Show that A is…

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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 &amp; T2 :X→X  are self -adjoint…

Q: Let T be a linear operator on a finite-dimensional Prove that T is normal if and only if eomplex…

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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: Let TE B(H) be a self-adjoint operator on a Hilbert space H (a) Use the continuous functional…

A: Given that: T∈BH be a self-adjoint operator on a Hilbert space H. The objective is to show that: a…

Q: Let T be a linear operator on an inner product space V. Let U1= T + T* and U2 = TT*. Prove that U1 =…

A: If T is an operator in an inner product space V over the field F together with an inner product,…

Q: Let T be a linear operator on a real inner product space V, and define H: V ×V →R by H(x, y) =…

A: Given: T is a linear operator on V.(a) To show: H is a bilinear form.H is defined as,

Q: The upper limit topology on ℝ is defined by the basis ℬ = {(a,b]  s.t.  a < b for all a,b ∈ ℝ}.…

A: TRUE

Q: Assume that T is a linear operator on a complex (not necessarily finite-dimensional)inner product…

A: (a) Suppose T is self-adjoint. Then

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: Let V be a vector space on R of dimension 4, and T be a linear operator on V. In case that there…

A: i uave used the definition of diagonalisation

Q: Let T be a linear operator on an inner product space V, and suppose that ||T(x)||= ||x|| for all x.…

A: According to property of norm the value of ||x||=0 iff x=0.