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Asked Mar 4, 2020
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Let V be a finite-dimensional inner product space over F.
(a) Parseval's Identity. Let {v1, v2, ..., vn} be an orthonormal basis for V. For any x, y e V prove that
(π, 1) Σ(α, υ) U,υ.
%3D
(b) Use (a) to prove that if B is an orthonormal basis for V with inner
product < ·, >, then for any x, y E V
< o B(x), PB(Yy)>' = <[x]B, [y]B>' = <x, y>,
where <·,> is the standard inner product on Fn.
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Let V be a finite-dimensional inner product space over F. (a) Parseval's Identity. Let {v1, v2, ..., vn} be an orthonormal basis for V. For any x, y e V prove that (π, 1) Σ(α, υ) U,υ. %3D (b) Use (a) to prove that if B is an orthonormal basis for V with inner product < ·, >, then for any x, y E V < o B(x), PB(Yy)>' = <[x]B, [y]B>' = <x, y>, where <·,> is the standard inner product on Fn.

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