Page 286 - Advanced Linear Algebra
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270 Advanced Linear Algebra
However, in general metric vector spaces, an orthogonal complement may not
be a vector space complement. In fact, Example 11.3 shows that in some cases
:~ :. In other cases, for example, if # is degenerate, then º#» ~ = .
However, as we will see, the orthogonal complement of is a vector space
:
complement if and only if either the sum is correct, =~ : b : , or the
intersection is correct, : q : ~ ¸ ¹ . Note that the latter is equivalent to the
nonsingularity of .
:
Many nice properties of orthogonality in real inner product spaces do carry over
to nonsingular metric vector spaces. Moreover, the next result shows that the
restriction to nonsingular spaces is not that severe.
Theorem 11.7 Let be a metric vector space. Then
=
=~ rad ²= ³ p :
where is nonsingular and rad ² = ³ is totally singular.
:
Proof. If is any vector space complement of rad ² = ³ , then rad ² = ³ : and so
:
=~ rad ²= ³ p :
Also, is nonsingular since rad ² : ³ rad ² = . ³
:
Here are some properties of orthogonality in nonsingular metric vector spaces.
In particular, if either or is nonsingular, then the orthogonal complement of
:
=
: always has the expected dimension,
dim²: ³ ~ dim²= ³ c dim²:³
even if : is not well behaved with respect to its intersection with .
:
Theorem 11.8 Let be a subspace of a finite-dimensional metric vector space
:
= .
1) If either or is nonsingular, then
:
=
dim²:³ b dim²: ³ ~ dim²= ³
Hence, the following are equivalent:
)
a =~ : b :
)
b : is nonsingular, that is, : q : ~ ¸ ¹
c ) =~ : p : .
)
2 If is nonsingular, then
=
a : ) ~ :
b)rad²:³ ~ rad²: ³
c : ) is nonsingular if and only if : is nonsingular.
Proof. For part 1), the map ¢= ¦ : i of Theorem 11.6 is surjective and has
kernel : . Thus, the rank-plus-nullity theorem implies that
