Page 116 - Linear Algebra Done Right

Norms
103
Proof: Suppose that u, v are orthogonal vectors in V. Then
The proof of the
Pythagorean theorem
2
u + v = u + v, u + v
shows that 6.4 holds if
2 2
= u + v + u, v + v, u and only if
2
2
= u + v , u, v + v, u , which
equals 2Re u, v ,is 0.
as desired.
Thus the converse of
the Pythagorean
Suppose u, v ∈ V. We would like to write u as a scalar multiple of v
theorem holds in real
plus a vector w orthogonal to v, as suggested in the next picture.
inner-product spaces.
u
w
λv
v

0
An orthogonal decomposition

To discover how to write u as a scalar multiple of v plus a vector or-
thogonal to v, let a ∈ F denote a scalar. Then

u = av + (u − av).
Thus we need to choose a so that v is orthogonal to (u−av). In other
words, we want
2
0 = u − av, v = u, v − a v .
2
The equation above shows that we should choose a to be u, v / v
(assume that v = 0 to avoid division by 0). Making this choice of a,we
can write

u, v u, v
6.5 u = v + u − v .
v 2 v 2
As you should verify, if v = 0 then the equation above writes u as a
scalar multiple of v plus a vector orthogonal to v.
The equation above will be used in the proof of the next theorem,
which gives one of the most important inequalities in mathematics.