Linear Algebra, 2020a

Section II. Linear Geometry 45
and rewriting
0 |⃗u | 2 |⃗v | 2 − 2 ( |⃗v | ⃗u ) • ( |⃗u |⃗v )+|⃗u | 2 |⃗v | 2
is true. But factoring shows that it is true
0 ( |⃗u |⃗v − |⃗v | ⃗u ) • ( |⃗u |⃗v − |⃗v | ⃗u )
since it only says that the square of the length of the vector |⃗u |⃗v − |⃗v | ⃗u is not
negative. As for equality, it holds when, and only when, |⃗u |⃗v − |⃗v | ⃗u is ⃗0. The
check that |⃗u |⃗v = |⃗v | ⃗u if and only if one vector is a nonnegative real scalar
multiple of the other is easy.
QED
This result supports the intuition that even in higher-dimensional spaces,
lines are straight and planes are flat. We can easily check from the definition
that linear surfaces have the property that for any two points in that surface,
the line segment between them is contained in that surface. But if the linear
surface were not flat then that would allow for a shortcut.
P
Q
Because the Triangle Inequality says that in any R n the shortest cut between
two endpoints is simply the line segment connecting them, linear surfaces have
no bends.
Back to the definition of angle measure. The heart of the Triangle Inequality’s
proof is the ⃗u • ⃗v |⃗u ||⃗v | line. We might wonder if some pairs of vectors satisfy
the inequality in this way: while ⃗u • ⃗v is a large number, with absolute value
bigger than the right-hand side, it is a negative large number. The next result
says that does not happen.
2.6 Corollary (Cauchy-Schwarz Inequality) For any ⃗u,⃗v ∈ R n ,
| ⃗u • ⃗v | | ⃗u ||⃗v |
with equality if and only if one vector is a scalar multiple of the other.
Proof The Triangle Inequality’s proof shows that ⃗u • ⃗v |⃗u ||⃗v | so if ⃗u • ⃗v is
positive or zero then we are done. If ⃗u • ⃗v is negative then this holds.
| ⃗u • ⃗v | =−(⃗u • ⃗v )=(−⃗u ) • ⃗v |−⃗u ||⃗v | = |⃗u ||⃗v |
The equality condition is Exercise 19.
QED
The Cauchy-Schwarz inequality assures us that the next definition makes
sense because the fraction has absolute value less than or equal to one.