Linear Algebra and Its Applications · Problem 3.2.1

Orthogonality — Gilbert Strang

Mathematics · cosines-projections-onto-lines

Problem

(a) Given any two positive numbers

x

and

y

, choose the vector

b

equal to

(\sqrt{x}, \sqrt{y})

, and choose

a = (\sqrt{y}, \sqrt{x})

. Apply the Schwarz inequality to compare the arithmetic mean

\frac{1}{2}(x+y)

with the geometric mean

\sqrt{xy}

. (b) Suppose we start with a vector from the origin to the point

x

, and then add a vector of length

\|y\|

connecting

x

x+y

. The third side of the triangle goes from the origin to

x+y

. The triangle inequality asserts that this distance cannot be greater than the sum of the first two:

\|x+y\| \leq \|x\| + \|y\|.

After squaring both sides, and expanding

(x+y)^T(x+y)

, reduce this to the Schwarz inequality.