Let T be a linear operator on a real inner product space V, and define H: V ×V →R by H(x, y) = <x,T(y)>for all x, y ∈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.

Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.4: Linear Transformations
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Let T be a linear operator on a real inner product space V, and define H: V ×V →R by H(x, y) = <x,T(y)>for all x, y ∈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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