Page 54 - Advanced Linear Algebra
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38 Advanced Linear Algebra
: is closed under linear combinations, that is,
Á -Á "Á # : ¬ " b # :
Example 1.2 Consider the vector space =² Á ³ of all binary -tuples, that is,
-tuples of 's and 's. The weight M ² # ³ # of a vector = ² Á ³ is the number
#
of nonzero coordinates in . For instance, M ² ³ ~ . Let , be the set of
all vectors in of even weight. Then , is a subspace of ² = Á . ³
=
To see this, note that
M M²" b #³ ~ ²"³ M ²#³ c M b ²" q #³
where "q# is the vector in = ² Á ³ whose th component is the product of the
#
"th components of and , that is,
²" q #³ ~ " h #
Hence, if M and M²"³ ²#³ are both even, so is M ²" b #³ . Finally, scalar
multiplication over - is trivial and so , is a subspace of ² = Á ³ , known as
the even weight subspace of =² Á ³ .
Example 1.3 Any subspace of the vector space =² Á ³ is called a linear code .
Linear codes are among the most important and most studied types of codes,
because their structure allows for efficient encoding and decoding of
information.
The Lattice of Subspaces
The set I²= ³ of all subspaces of a vector space is partially ordered by set
=
I
inclusion. The zero subspace ¸ ¹ is the smallest element in ²= ³ and the entire
space is the largest element.
=
=
If :Á ; ²= ³ , then : q ; is the largest subspace of that is contained in
I
;
both and . In terms of set inclusion, q : ; is the greatest lower bound of :
:
and :
;
:q ; ~ glb ¸:Á ;¹
Similarly, if ¸: 2¹ is any collection of subspaces of = , then their
intersection is the greatest lower bound of the subspaces:
:~ glb ¸: 2¹
2
)
(
I
On the other hand, if :Á ; ²= ³ and is infinite , then : r ; ²= ³ if
-
I
and only if : ; or ; : . Thus, the union of two subspaces is never a
subspace in any “interesting” case. We also have the following.