Page 115 - Linear Algebra Done Right

Chapter 6. Inner-Product Spaces
102
Norms
For v ∈ V, we define the norm of v, denoted v ,by

v = v, v .
For example, if (z 1 ,...,z n ) ∈ F n (with the Euclidean inner product),
then

2
2
(z 1 ,...,z n ) = |z 1 | +· · ·+|z n | .
As another example, if p ∈P m (F) (with inner product given by 6.2),
then

1
2
p = |p(x)| dx.
0
Note that v = 0 if and only if v = 0 (because v, v = 0 if and only
if v = 0). Another easy property of the norm is that av =|a| v
for all a ∈ F and all v ∈ V. Here’s the proof:

2
av = av, av
= a v, av
= a¯ a v, v
2
2
=|a| v ;
taking square roots now gives the desired equality. This proof illus-
trates a general principle: working with norms squared is usually easier
than working directly with norms.
Some mathematicians Two vectors u, v ∈ V are said to be orthogonal if u, v = 0. Note
use the term that the order of the vectors does not matter because u, v = 0if
perpendicular, which and only if v, u = 0. Instead of saying that u and v are orthogonal,
means the same as sometimes we say that u is orthogonal to v. Clearly 0 is orthogonal
orthogonal. to every vector. Furthermore, 0 is the only vector that is orthogonal to
itself.
2
The word orthogonal For the special case where V = R , the next theorem is over 2,500
comes from the Greek years old.
word orthogonios,
which means 6.3 Pythagorean Theorem: If u, v are orthogonal vectors in V, then
right-angled.
2
2
2
6.4 u + v = u + v .