Linear Algebra and Its Applications · Problem 3.2.6

Orthogonality — Gilbert Strang

Mathematics · cosines-projections-onto-lines

Problem

The Schwarz inequality has a one-line proof if

a

and

b

are normalized ahead of time to be unit vectors:

|a^Tb| = \left|\sum a_jb_j\right| \leq \sum|a_j||b_j| \leq \sum\frac{|a_j|^2+|b_j|^2}{2} = \frac{1}{2}+\frac{1}{2} = \|a\|\|b\|.

Which previous problem justifies the middle step?