Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I must derive the following relation using the Cauchy Schwarz inequality for any collection of $N$ real numbers $a_1,a_2,...,a_N$:

$$\left(\frac{a_1+a_2+\cdots+a_N}{N}\right)^2\leq\frac{a^2_1+a^2_2+\cdots+a^2_N}{N}.$$

This says that the square of the average is less or equal than the average of the squares.

I must consider $v \in\mathbb{R}^N$ with components given by the numbers $a_1,a_2,...,a_N$ and find a suitable $w \in\mathbb{R}^N$ such that $v \cdot w=\dfrac{a_1+a_2+\cdots+a_N}{N}$

Some guidance and direction would be greatly appreciated.

share|improve this question
    
$w:= (1/N, 1/N, \ldots, 1/N)$. –  njguliyev Oct 23 '13 at 13:50
    
Ok that works. I'm using $|v \cdot w|^2 ≤ ||v||^2 ||w||^2$ and so acquiring the LHS is obvious. What would I need to do to derive the relation? –  Harry R Oct 23 '13 at 13:55
    
Ignore that. I now have $v \cdot w$, $v \cdot v$ and $w \cdot w$. –  Harry R Oct 23 '13 at 14:06

1 Answer 1

Hint: proceed with the vectors $(a_1,a_2,\cdots,a_n)$ and $(1,1,\cdots,1)$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.