Page 209 - A Course in Linear Algebra with Applications

Chapter Seven

ORTHOGONALITY IN
VECTOR SPACES

The notion of two lines being perpendicular, or orthogo-
nal, is very familiar from analytical geometry. In this chapter
we show how to extend the elementary concept of orthogonal-
ity to abstract vector spaces over R or C. Orthogonality turns
out to be a tool of extraordinary utility with many applica-
tions, one of the most useful being the well-known Method
n
of Least Squares. We begin with R , showing how to define
orthogonality in this vector space in a way which naturally
generalizes our intuitive notion of perpendicularity in three-
dimensional space.

7.1 Scalar Products in Euclidean Space

n
Let X and Y be two vectors in R , with entries x\,..., x n
and 2/1,..., y n respectively. Then the scalar product of X and
Y is defined to be the matrix product

(Vi\
V2
T
X Y = (x 1x 2 ... x n) = Xiyi + X2IJ2 H V XnVn-

T
T
This is a real number. Notice that X Y = Y X, so the scalar
product is symmetric in X and Y. Of particular interest is the
scalar product of X with itself

T
2
X X = xl + xl + --- + x n.
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