Page 60 - Advanced Linear Algebra

44 Advanced Linear Algebra

Then 2) implies that ~ ~ and ~! " " for all "~ Á Ã Á . Hence, 2)
implies 3).

Finally, suppose that 3) holds. If

p s
£#: q :

q £ t
and
then #~ :

~ b Ä b

are nonzero. But this violates 3).
where  :
can be written in the form
Example 1.5 Any matrix ( C
! !
( ~ ²( b( ³b ²( c( ³ ~ ) b* (1.1 )

where ( ! is the transpose of . It is easy to verify that is symmetric and is
(
*
)
)
skew-symmetric and so 1.1 is a decomposition of as the sum of a symmetric
(
(
matrix and a skew-symmetric matrix.
Since the sets Sym and SkewSym of all symmetric and skew-symmetric
matrices in C are subspaces of C , we have
C ~ b Sym SkewSym
Z
Z
;
Furthermore, if :b ; ~ : b ; Z , where and are symmetric and and ; Z
:
:
are skew-symmetric, then the matrix
Z
Z
<~ : c : ~ ; c ;
is both symmetric and skew-symmetric. Hence, provided that char²-³ £ , we
must have <~ and so : ~ : Z and ;~ ; Z . Thus,
C ~ l Sym SkewSym
Spanning Sets and Linear Independence
A set of vectors spans a vector space if every vector can be written as a linear
combination of some of the vectors in that set. Here is the formal definition.

(
Definition The subspace spanned or subspace generated) by a nonempty set
: = of vectors in is the set of all linear combinations of vectors from :
:
º:» ~ span ²:³ ~ ¸ # b Ä b # “  -Á #  :¹