Page 53 - Advanced Linear Algebra
P. 53
Vector Spaces 37
² Á Ã Á ³ ~ ² Á Ã Á ³
When convenient, we will also write the elements of - in column form.
- - with elements, we write ² = Á ³ for - .
When is a finite field
)
4 Many sequence spaces are vector spaces. The set Seq²-³ of all infinite
sequences with members from a field - is a vector space under the
componentwise operations
² ³ b ²! ³ ~ ² b ! ³
and
² ³ ~ ² ³
of all sequences of complex numbers that
In a similar way, the set
converge to is a vector space, as is the set M B of all bounded complex
sequences. Also, if is a positive integer, then the set of all complex
M
sequences ² ³ for which
B
(( B
~
is a vector space under componentwise operations. To see that addition is a
binary operation on , one verifies Minkowski's inequality
M
° ° °
B B B
b ! ( ( ( ( b ! (
8 ( 9 8 9 8 9
~ ~ ~
which we will not do here.
Subspaces
Most algebraic structures contain substructures, and vector spaces are no
exception.
Definition A subspace of a vector space is a subset of that is a vector
=
=
:
space in its own right under the operations obtained by restricting the
:
:
operations of = to . We use the notation : = to indicate that is a
=
=
subspace of and : = to indicate that is a proper subspace of , that is,
:
: = but : £ = . The zero subspace of = is ¸ ¹.
Since many of the properties of addition and scalar multiplication hold a fortiori
in a nonempty subset , we can establish that is a subspace merely by
:
:
checking that is closed under the operations of .
:
=
Theorem 1.1 A nonempty subset of a vector space is a subspace of if
=
:
=
and only if is closed under addition and scalar multiplication or, equivalently,
:
