MA 265 LECTURE NOTES: WEDNESDAY, FEBRUARY 20
Subspaces
Definition.
Let (
V,
⊕
,
⊙
) be a real vector space. We say that a subset
W
⊆
V
is a
subspace
of
V
if
(1)
W
negationslash
=
∅
, and
(2) (
W,
⊕
,
⊙
) is a real vector space.
We spend the rest of the lecture giving examples and discussing relevant results.
Determining Subspaces.
Let (
V,
⊕
,
⊙
) be a real vector space, and
W
⊆
V
be a subset. We give a method
to determine when
W
is a subspace. We will show that
W
is a subspace of
V
if and only if
(i)
W
is nonempty:
W
negationslash
=
∅
.
(ii)
W
is closed under
⊕
:
If
u
,
v
∈
W
then
u
⊕
v
∈
W
.
(iii)
W
is closed under
⊙
:
If
c
∈
R
and
u
∈
W
then
c
⊙
u
∈
W
.
Hence in order to check that a subset is a subspace, it suffices to check that
W
is nonempty, closed under
vector addition, and closed under scalar multiplication.
We explain why this is true. First assume that
W
is a subspace of
V
. Then (
W,
⊕
,
⊙
) is a real vector
space. This means
u
⊕
v
∈
W
for all
u
,
v
∈
W
so that (ii) holds; and
c
⊙
u
∈
W
for all
c
∈
R
and
u
∈
W
so that (iii) holds. Since
0
∈
W
this shows that (i) holds.
Now assume that properties (i), (ii), and (iii) hold. We must show that
(1)
W
negationslash
=
∅
, and
(2) (
W,
⊕
,
⊙
) is a real vector space. That is, if
u
,
v
,
w
∈
W
and
c, d
∈
R
then we must show that the
following properties are valid:
•
(
Commutativity:
)
u
⊕
v
=
v
⊕
u
.
•
(
Associativity
:)
u
⊕
(
v
⊕
w
) = (
u
⊕
v
)
⊕
w
, and
c
⊙
(
d
⊙
u
) = (
cd
)
⊙
u
.
•
(
Distributivity:
)
c
⊙
(
u
⊕
v
) = (
c
⊙
u
)
⊕
(
c
⊙
v
), and (
c
+
d
)
⊙
u
= (
c
⊙
u
)
⊕
(
d
⊙
u
).
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 Spring '10
 ...
 Linear Algebra, Vector Space, real vector space

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