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Asked Mar 4, 2020
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Let T be a linear operator on a real inner product space V, and define H: V x V → R by H(x, y) = <x,T(y)> for all x, y e V.
(a) Prove that H is a bilinear form.
(b) Prove that H is symmetric if and only if T is self-adjoint.
(c) What properties must T have for H to be an inner product on V?
(d) Explain why H may fail to be a bilinear form if V is a complex inner product space.
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Let T be a linear operator on a real inner product space V, and define H: V x V → R by H(x, y) = <x,T(y)> for all x, y e V. (a) Prove that H is a bilinear form. (b) Prove that H is symmetric if and only if T is self-adjoint. (c) What properties must T have for H to be an inner product on V? (d) Explain why H may fail to be a bilinear form if V is a complex inner product space.

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