Page 282 - Advanced Linear Algebra
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266 Advanced Linear Algebra
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1 A vector #= is degenerate if #= . The set = of all degenerate
vectors is called the radical of and denoted by rad ² = ³ . Thus,
=
rad²= ³ ~ =
)
2 = is nonsingular , or nondegenerate , if rad²= ³ ~ ¸ ¹ .
)
3 = is singular , or degenerate , if rad²= ³ £ ¸ ¹ .
4 = ) is totally singular , or totally degenerate , if rad = ² ³ ~ = .
Some of the above terminology is not entirely standard, so care should be
exercised in reading the literature.
Theorem 11.3 A metric vector space = is nonsingular if and only if all
representing matrices 4 8 are nonsingular.
A note of caution is in order. If is a subspace of a metric vector space , then
=
:
rad²:³ denotes the set of vectors in : that are degenerate in :, that is, rad²:³ is
the radical of , as a metric vector space in its own right. However, : denotes
:
the set of all vectors in that are orthogonal to . Thus,
=
:
rad²:³ ~ : q :
Note also that
rad²:³ ~ : q : : q :~ rad²: ³
and so if is singular, then so is : .
:
Example 11.3 Recall that =² Á ³ is the set of all ordered -tuples whose
. (See Example 11.2.) It is easy to see
components come from the finite field -
that the subspace
: ~ ¸ Á Á Á ¹
of = ² Á ³ has the property that : ~ : . Note also that = ² Á ³ is nonsingular
and yet the subspace is totally singular.
:
The following result explains why we restrict attention to symmetric or alternate
forms (which includes skew-symmetric forms).
Theorem 11.4 Let be a vector space with a bilinear form. The following are
=
equivalent:
1 Orthogonality is a symmetric relation, that is,
)
%& ¬ & %
2 The form on = ) is symmetric or alternate, that is, = is a metric vector
space.
