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.
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