MATH 294
Chapter 3.2 Subspaces of
R
n
; Bases and Linear Independence
Ex 3.2.5:
Give a geometrical description of all subspaces of
R
3
.
Answer:
Keep in mind that any subspace must
contain origin. So subspaces are
{
0
}
, lines, and planes containing the origin. And don’t forget
R
3
itself.
Ex 3.2.6:
Consider two subspaces V and W of
R
n
.
a.)
Is the intersection
U
=
V
W
necessarily a subspace
of
R
n
?
Solution:
Both V and W are subspaces. Then they contain 0. So, U contains 0 (as U is the intersection
of V and W). If
x
and
y
are in U then both
x
and
y
are in V and W at the same time. As V and W are subspaces
then
x
+
y
is in V and W then
x
+
y
is in the intersection of V and W, that is, in U. The same reasoning works for
checking
kx
∈
U
for any
x
∈
U
and any constant
k
∈
R
.
b.)
Is the union
U
=
V
W
necessarily a subspace of
R
n
.
No, it is not.
Because if
x
∈
V
and
y
∈
W
then
x, y
∈
U
=
V
W
but
x
+
y
is not necessarily in U. Consider the following example.
Let n=3 (so U and W
are subspaces of
R
3
) and V= span
{
(1
,
0
,
0)
}
and W=span
{
(0
,
1
,
0)
}
then U contains all vectors in
R
3
having form
(
λ
1
,
0
,
0) or (0

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