Page 273 - Matrix Analysis & Applied Linear Algebra

CHAPTER 5

Norms,

Inner Products,

and Orthogonality

5.1 VECTOR NORMS

A significant portion of linear algebra is in fact geometric in nature because
much of the subject grew out of the need to generalize the basic geometry of
2 3
and to nonvisual higher-dimensional spaces. The usual approach is to
2 3
coordinatize geometric concepts in and , and then extend statements
n n
concerning ordered pairs and triples to ordered n-tuples in and C .
2 3
For example, the length of a vector u ∈ or v ∈ is obtained from
the Pythagorean theorem by computing the length of the hypotenuse of a right
triangle as shown in Figure 5.1.1.

u = (x,y) v = (x,y,z)
x 2 + y 2 x 2 + y 2 + z 2 z

||u|| = y ||v|| =

x
x
y
Figure 5.1.1
This measure of length,

2
2
2
2
u = x + y 2 and v = x + y + z ,