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Parseval's Identity:

Parseval's Identity:

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter5: Inner Product Spaces
Section5.3: Orthonormal Bases:gram-schmidt Process
Problem 17E: Complete Example 2 by verifying that {1,x,x2,x3} is an orthonormal basis for P3 with the inner...
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3. LADW #5.2.3: Let {v₁,..., Vn} be an orthonormal basis in V. (a) Prove that for any x = Σαν and y = Σβιve, we have that k=1 n (b) Deduce from this the Parseval's Identity: 72 n k=1 (x, y) = Σακβκ. k=1 n (x, y) = Σ(x, vk) (y, vk). k=1 Note: Part (a) shows us the usefulness of an orthonormal basis B. It says that we can compute the inner product of two vectors in V by computing the standard inner product in F (instead of in V) of the coordinate vectors with respect to B.
Transcribed Image Text:3. LADW #5.2.3: Let {v₁,..., Vn} be an orthonormal basis in V. (a) Prove that for any x = Σαν and y = Σβιve, we have that k=1 n (b) Deduce from this the Parseval's Identity: 72 n k=1 (x, y) = Σακβκ. k=1 n (x, y) = Σ(x, vk) (y, vk). k=1 Note: Part (a) shows us the usefulness of an orthonormal basis B. It says that we can compute the inner product of two vectors in V by computing the standard inner product in F (instead of in V) of the coordinate vectors with respect to B.
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