Exercise 3.2.6. Suppose that and are unit vectors. Then a one-line proof of the Schwarz inequality is as follows:

What previous exercise justifies the middle step of this proof?

Answer: From exercise 3.2.1(a) we have for any positive and . We can easily extend this result to any non-negative and : If then and as long as . So if and . A similar argument shows that if and . Combined with the previous result this shows that for all and .

Let and . We then have or . This is the middle step of the one-line proof of the Schwarz inequality shown above.

Note that exercise 3.2.1(a) actually used the Schwarz inequality to prove that , so strictly speaking the above proof is circular. To remedy this we can prove the result from exercise 3.2.1(a) without using the Schwarz inequality. The proof (adapted from Wikipedia) is as follows:

We know that for any and . Expanding the lefthand side we have . Substituting into the inequality we have or . Adding to both sides of the inequality we have or . If and then and we can take the square root of both sides of the inequality to obtain or .

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books.