Let V be a finite-dimensional inner product space, and let E be an idempotent linear operator on V. Prove that E is self-adjoint iff EE*=E*E.

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

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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 and U be self-adjoint operators on an inner product space. Prove that TU is self-adjoint if…

A: It is given that T and U are two self-adjoint operators on an inner product space.

Q: Let V be afinite dimensional inner product space and let E be an idempotent linear operator on V.…

A: Suppose E2=EE2=E. Then E has a basis of eigenvectors with eigenvalues 0 and 1 because every vector…

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

Q: Let V be an inner product space. Show that ||cu|| |c| ||u|| for any vector u and any scalar c.

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

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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 =…

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Q: Let V be an inner product space and T be a linear operator on V with || Tv||= ||v||  ∀ v ∈ V. Show…

A: Let V be an inner product space and T be a linear operator on V with Tv=v ∀v∈V.

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

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

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Q: Let L be a linear operator on a vector space V. Define Ln, n ≥ 1, recursively by L1 = L Lk+1(v) = L…

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

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A: TRUE

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Q: Let T: V→W be a linear transformation with vector spaces V and W both of dimension n. Then T is…

A: True

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.

Q: Let V be a finite-dimensional inner product space, and let T be a linear operator on V. Prove that…

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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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