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Monday January 26
−
Lecture 10 :
Subspaces of R
n
(Refers to section 3.4)
Objectives:
1.
Define
subspace
of a vector space
R
n
.
2.
Prove that the intersection of two subspaces is a subspace.
3.
Prove that the span of a finite set of vectors in
R
n
is a subspace of
R
n
.
10.1
Introduction – Up to this point in the course we have referred to the elements of the
set
R
n
as vectors.
•
For reasons which we will make clear later we will now refer to
R
n
not simply as
a set but as a
vector space
.
•
For example, rather than say the set
R
3
we will say the
vector space
R
3
.
10.2
Definition – We say that a
subset
W
of the vector space
R
n
is a
subspace
of
R
n
if
•
W
contains the vector
0
and,
•
W
is closed under linear combinations.
That means, if
v
and
w
belong to
W
then
so does
α
v
+
β
w
for any
α
and
β
.
10.3
Observation – The vector space
R
n
is a subspace of itself, since it satisfies all the 2
conditions required to be a subspace.
10.4
Examples
10.4.1
Let
V
=
R
2
. Is
W
= [0,1]
2
= { (
x, y
) : 0
≤
x
≤
1
,
0
≤
y
≤
1} a subspace of
R
2
?
Answer: No! It is NOT closed under addition. Both (1, 1/2) and (1/2, 1) belong to
W
,
but (1, 1/2) + (1/2, 1) = (3/2, 3/2) is not in
W
.
10.4.2
Example : Is
W
= { (
x, y, z
) in
R
3
:
x + y + z
= 0} a subspace of
R
3
? Yes, since
o
W
contains (0,0,0).

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o
Suppose
v
= (
a, b, c
) and
w
= (
d, e, f
) are in
W
. Now
W
is all vectors
x
in
R
3
such that <
n
,
x
> = <(1, 1, 1),
x
> = 0.
o

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