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Question 14. Let -0-0-0) U2 = = 1 3√2 1 3√/2 4 1 √2 √2 3√2, (b) Let x = (1, 1, 1)T. Write x as a linear combination of u₁, U₂, and u3 using Theorem 21.13 and use Parseval's formula to compute ||x||.

Question 14. Let -0-0-0) U2 = = 1 3√2 1 3√/2 4 1 √2 √2 3√2, (b) Let x = (1, 1, 1)T. Write x as a linear combination of u₁, U₂, and u3 using Theorem 21.13 and use Parseval's formula to compute ||x||.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question 14. Let 1 3√2 1 -@)-0- 3√2 4 3√2 = 1 3 U3 = 2 0 (b) Let x = (1, 1, 1)T. Write x as a linear combination of u₁, u2, and u3 using Theorem 21.13 and use Parseval's formula to compute ||x||.
Transcribed Image Text:Question 14. Let 1 3√2 1 -@)-0- 3√2 4 3√2 = 1 3 U3 = 2 0 (b) Let x = (1, 1, 1)T. Write x as a linear combination of u₁, u2, and u3 using Theorem 21.13 and use Parseval's formula to compute ||x||.
Theorem 21.13 (Coordinate w.r.t orthonormal basis) Let {u₁,,um} be the orthonormal basis for the inner product vector space V, and for any v € V, v can be decomposed as B = m Σ < vu; > u; i=1 Proof. For any v E V, it can be written as a linear combination of the orthonormal basis as follows: V = C₁U₁ + + Cmum Taking the inner product with u; on both sides of the above equation, one has: < v, u¡ >= c; || u; ||²= c; since u; (i = 1, ... , m) are the unit vectors.
Transcribed Image Text:Theorem 21.13 (Coordinate w.r.t orthonormal basis) Let {u₁,,um} be the orthonormal basis for the inner product vector space V, and for any v € V, v can be decomposed as B = m Σ < vu; > u; i=1 Proof. For any v E V, it can be written as a linear combination of the orthonormal basis as follows: V = C₁U₁ + + Cmum Taking the inner product with u; on both sides of the above equation, one has: < v, u¡ >= c; || u; ||²= c; since u; (i = 1, ... , m) are the unit vectors.
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