Page 220 - A Course in Linear Algebra with Applications

204 Chapter Seven: Orthogonality in Vector Spaces

Because of 7.1.5 it is meaningful to define the angle between
n
two non-zero vectors X and Y in R to be the angle 9 in the
interval [0, ir] such that

T
X Y

An important consequence of 7.1.5 is

Theorem 7.1.6 (The Triangle Inequality)
n
If X and Y are vectors in R , then
\\X + Y\\ < ||X|| + ||y||.

Proof
Let the entries of X and Y be x\,... ,x n and y±,... ,y n re-
spectively. Then

T
T
T
r
||X + || 2 = (X + Yf{X + Y) = X X + X Y + Y X + YY T
T
T
and, since X Y = Y X, this equals
T
||X|| 2 + ||y|| 2 + 2X F.
T
By the Cauchy-Schwartz Inequality \X Y\ < \\X\\ \\Y\\, so it
follows that

2
\\X + Y\\ 2 < \\X\\ 2 + \\Y\\ 2 + 2\\X\\ \\Y\\ = (\\X\\ + ||F||) ,

which yields the desired inequality.