Page 234 - A Course in Linear Algebra with Applications

218 Chapter Seven: Orthogonality in Vector Spaces

mind as our model is the simple situation in three-dimensional
space where the orthogonal complement of a plane is the set
of line segments perpendicular to it.
Let 5 be a subspace of a real inner product space V.
The orthogonal complement of S is defined to be the set of all
vectors in V that are orthogonal to every vector in S: it is
denoted by the symbol
±
S .

Example 7.2.10
3
Let S be the subspace of R consisting of all vectors of the
form

(S)

where a and b are real numbers. Thus elements of S corre-
1
spond to line segments in the xy-plane. Equally clearly S - is
the set of all vectors of the form

( • ) •
These correspond to line segments along the 2-axis, hardly a
surprising conclusion.
The most fundamental property of an orthogonal com-
plement is that it is a subspace.

Theorem 7.2.3
Let S be a subspace of a real inner product space V. Then
1
(a) S - is a subspace of V;
±
(b) SnS = 0;
(c) if S is finitely generated, a vector v belongs to S 1 if
and only if it is orthogonal to every vector in some set of
generators of S.