Page 219 - A Course in Linear Algebra with Applications

7.1: Scalar Products in Euclidean Space 2Uo

parallelogram equals

(IQ sin 9)IP = \\X\\ \\Y\\ sin 6 = \\X x Y\\.

Q

Orthogonality in R n
3
Having gained some insight from R , we are now ready
to define orthogonality in n-dimensional Euclidean space.
n
Let X and Y be two vectors in R . Then X and Y are
said to be orthogonal if

T
X Y = 0.
3
This a natural extension of orthogonality in R . It follows
from the definition that the zero vector is orthogonal to every
vector in R n and that no non-zero vector can be orthogonal
2
T
to itself: indeed X X = x\ + x\ + • • • + x n > 0 if X ^ 0.
n
It turns out that the inequality of 7.1.2 is valid for R .
Theorem 7.1.5 (Cauchy - Schwartz Inequality)
n
If X and Y are vectors in R , then
T
\X Y\ < 11X11 iiyii.

We shall not prove 7.1.5 at this stage since a more general
fact will be established in 7.2: see however Exercise 7.1.10.