Q42E TRUE OR FALSE42. If 鈥泪蹿 x1,x2,... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 5: Q42E (page 264)

TRUE OR FALSE
42. If 鈥泪蹿 $x_{1}, x_{2}, . .., x_{n}$ are any real numbers then the inequality
${({鈭�}_{k = 1}^{n} x_{i})}^{2} 鈮� n {鈭�}_{k = 1}^{n} (x_{k}^{2})$
must hold.

Short Answer

Expert verified

The given statement is true.

Step by step solution

Definition of Cauchy-Schwartz inequality

The Cauchy-Schwartz inequality is, $(\overset{鈫�}{x} 鈰� \overset{鈫�}{y}) 鈮� 鈥� \overset{鈫�}{x} 鈥� 鈥� \overset{鈫�}{y} 鈥�$ .

Explanation of the solution

Consider the statement as follows:

鈥泪蹿 $x_{1}, x_{2}, ..., x_{n}$ are any real numbers, then the inequality $({鈭�}_{k = 1}^{n} x_{k}) 鈮� {鈭�}_{k = 1}^{n} (x_{k}^{2})$ 鈥�.

Let $\overset{鈫�}{x} = [\begin{matrix} x_{1} \\ 鈰� \\ x_{n} \end{matrix}]$ and $\overset{鈫�}{y} = [\begin{matrix} 1 \\ 鈰� \\ 1 \end{matrix}]$ .

Then $\overset{鈫�}{x} 鈰� \overset{鈫�}{y}$ as follows.

$\begin{matrix} \overset{鈫�}{x} 鈰� \overset{鈫�}{y} = [\begin{matrix} x_{1} \\ 鈰� \\ x_{n} \end{matrix}] [\begin{matrix} 1 \\ 鈰� \\ 1 \end{matrix}] \\ = {鈭�}_{i = 1}^{n} x_{i} \end{matrix}$

Also ${鈥� \overset{鈫�}{x} 鈥�}^{2} = {鈭�}_{i = 1}^{n} x_{i}^{2}$ and ${鈥� \overset{鈫�}{y} 鈥�}^{2} = n$

Using the Cauchy-Schwartz inequality in the above as follows.

$\begin{matrix} (\overset{鈫�}{x} 鈰� \overset{鈫�}{y}) 鈮� 鈥� \overset{鈫�}{x} 鈥� 鈥� \overset{鈫�}{y} 鈥� \\ {({鈭�}_{i = 1}^{n} x_{i})}^{2} 鈮� n {鈭�}_{i = 1}^{n} (x_{i}^{2}) \end{matrix}$

Thus, the given statement 鈥泪蹿 $x_{1}, x_{2}, ..., x_{n}$ are any real numbers, then the inequality ${({鈭�}_{k = 1}^{n} x_{k})}^{2} 鈮� {鈭�}_{k = 1}^{n} (x_{k}^{2})$ 鈥� is true.

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91影视

TRUE OR FALSE
42. If 鈥泪蹿 $x_{1}, x_{2}, . .., x_{n}$ are any real numbers then the inequality
${({鈭�}_{k = 1}^{n} x_{i})}^{2} 鈮� n {鈭�}_{k = 1}^{n} (x_{k}^{2})$
must hold.

Short Answer

Step by step solution

Definition of Cauchy-Schwartz inequality

Explanation of the solution

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91影视

TRUE OR FALSE42. If 鈥泪蹿 x1,x2,...,xnare any real numbers then the inequality(鈭�k=1nxi)2鈮�n鈭�k=1n(xk2)must hold.

Short Answer

Step by step solution

Definition of Cauchy-Schwartz inequality

Explanation of the solution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Pure Maths

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Study anywhere. Anytime. Across all devices.

TRUE OR FALSE
42. If 鈥泪蹿 $x_{1}, x_{2}, . .., x_{n}$ are any real numbers then the inequality
${({鈭�}_{k = 1}^{n} x_{i})}^{2} 鈮� n {鈭�}_{k = 1}^{n} (x_{k}^{2})$
must hold.