Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 5: Q 5.5-25E (page 262)

Question: Find the norm || $\overset{鈫�}{x}$ || of
$\overset{鈫�}{x} = (1, \frac{1}{2}, \frac{1}{3}, . ...., \frac{1}{n} . ....)$ in I₂.
( I₂ is defined in Example 2.)

Short Answer

Expert verified

The norm will be $| | \overset{鈫�}{x} | | = \frac{蟺}{6}$

Step by step solution

Consider the theorem below.

$a_{0}^{2} + b_{1}^{2} + c_{1}^{2} + . ... + b_{n}^{2} + c_{n}^{2} + . ... = | | f | |^{2}$

The infinite series of the squares of the Fourier coefficients of a piecewise continuous function fconverges to $| | f | |^{2}$ .

Consider the function f(t)=t , $鈭� 蟺鈮� t 鈮� 蟺$ . Observe that, a₀ and c_k will be zero for every k as f(t)=t

Is an odd function. So,

$\begin{matrix} b_{k} = 鈱�f, \sin (k t)鈱� \\ = \frac{1}{蟺} {鈭�}_{鈭� 蟺}^{蟺} t \sin (k t) d t \\ = \frac{2}{蟺} {鈭�}_{0}^{蟺} t \sin (k t) d t \\ = \frac{2}{蟺} {[鈭� \frac{t \cos (t)}{k}]}_{0}^{蟺} + {鈭�}_{0}^{蟺} \frac{\cos (t)}{k} d t \\ = \frac{2}{蟺} {[鈭� \frac{蟺 \cos (k 蟺)}{k} + \frac{\sin (k t)}{k^{2}}]}_{0}^{蟺} \\ = 鈭� \frac{2}{k} \cos (k 蟺) \end{matrix}$

Thus, we have

$\begin{matrix} 4 + \frac{4}{2^{2}} + \frac{4}{3^{2}} + \frac{4}{4^{2}} + .... + = | | f | |^{2} \\ = \frac{1}{蟺} {鈭�}_{鈭� 蟺}^{蟺} t^{2} d t \\ = \frac{2}{蟺} \frac{蟺^{3}}{3} \\ 鈬� 1 + \frac{1}{2^{2}} + \frac{1}{3^{2}} + .... + = \frac{蟺^{2}}{6} \end{matrix}$

Hence, the norm is $| | \overset{鈫�}{x} | | = \frac{蟺}{6}$ .

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91影视

Question: Find the norm || $\overset{鈫�}{x}$ || of
$\overset{鈫�}{x} = (1, \frac{1}{2}, \frac{1}{3}, . ...., \frac{1}{n} . ....)$ in I₂.
( I₂ is defined in Example 2.)

Short Answer

Step by step solution

Consider the theorem below.

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91影视

Question: Find the norm || x鈫�|| ofx鈫�=(1,12,13,.....,1n.....)in I2.( I2 is defined in Example 2.)

Short Answer

Step by step solution

Consider the theorem below.

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Discrete Mathematics

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Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Question: Find the norm || $\overset{鈫�}{x}$ || of
$\overset{鈫�}{x} = (1, \frac{1}{2}, \frac{1}{3}, . ...., \frac{1}{n} . ....)$ in I₂.
( I₂ is defined in Example 2.)