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||.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.2: Trigonometric Equations
Problem 91E
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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||.
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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