My Favorite One-Liners: Part 100

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

Today’s quip is one that I’ll use surprisingly often:

If you ever meet a mathematician at a bar, ask him or her, “What is your favorite application of the Cauchy-Schwartz inequality?”

The point is that the Cauchy-Schwartz inequality arises surprisingly often in the undergraduate mathematics curriculum, and so I make a point to highlight it when I use it. For example, off the top of my head:

1. In trigonometry, the Cauchy-Schwartz inequality states that

$|{\bf u} \cdot {\bf v}| \le \; \parallel \!\! {\bf u} \!\! \parallel \cdot \parallel \!\! {\bf v} \!\! \parallel$

for all vectors ${\bf u}$ and ${\bf v}$ . Consequently,

$-1 \le \displaystyle \frac{ {\bf u} \cdot {\bf v} } {\parallel \!\! {\bf u} \!\! \parallel \cdot \parallel \!\! {\bf v} \!\! \parallel} \le 1$ ,

which means that the angle

$\theta = \cos^{-1} \left( \displaystyle \frac{ {\bf u} \cdot {\bf v} } {\parallel \!\! {\bf u} \!\! \parallel \cdot \parallel \!\! {\bf v} \!\! \parallel} \right)$

is defined. This is the measure of the angle between the two vectors ${\bf u}$ and ${\bf v}$ .

2. In probability and statistics, the standard deviation of a random variable $X$ is defined as

$\hbox{SD}(X) = \sqrt{E(X^2) - [E(X)]^2}$ .

The Cauchy-Schwartz inequality assures that the quantity under the square root is nonnegative, so that the standard deviation is actually defined. Also, the Cauchy-Schwartz inequality can be used to show that $\hbox{SD}(X) = 0$ implies that $X$ is a constant almost surely.

3. Also in probability and statistics, the correlation between two random variables $X$ and $Y$ must satisfy

$-1 \le \hbox{Corr}(X,Y) \le 1$ .

Furthermore, if $\hbox{Corr}(X,Y)=1$ , then $Y= aX +b$ for some constants $a$ and $b$ , where $a > 0$ . On the other hand, if $\hbox{Corr}(X,Y)=-1$ , if $\hbox{Corr}(X,Y)=1$ , then $Y= aX +b$ for some constants $a$ and $b$ , where $a < 0$ .

Since I’m a mathematician, I guess my favorite application of the Cauchy-Schwartz inequality appears in my first professional article, where the inequality was used to confirm some new bounds that I derived with my graduate adviser.

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My Favorite One-Liners: Part 100

Published by John Quintanilla

2 thoughts on “My Favorite One-Liners: Part 100”

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Published by John Quintanilla

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