MATH1850U/2050U:
Chapter 4 cont.
..
1
GENERAL VECTOR SPACES cont.
..
Subspaces (4.2; pg. 179)
Definition:
A subset
W
of
V
is called a
subspace
of
V
if
W
is itself a vector space under the
addition and scalar multiplication defined on
V
.
Example:
Let
W
consist of the points on the line
1
x
y
and so,
W
is a subset of
R
2
.
Show
that
W
is NOT
a subspace of
R
2
.
Example:
The set of points
W
that lie on a line through the origin is a subset of
R
3
.
Show that
W
is a subspace of
R
3
.
Theorem (Subspaces):
If
W
is a set of one or more vectors from a vector space
V
, then
W
is a subspace of
V
if and only if the following conditions hold:
a)
If
u
and
v
are vectors in
W
, then
v
u
is in
W
b)
If
k
is any scalar and
u
is any vector in
W
, then
u
k
is in
W
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Chapter 4 cont.
..
2
Example:
The set of all points on a plane is a subset of
R
3
.
Every plane through the origin is a
subspace of
R
3
.
Example:
In the previous section, we saw that the set that contained only the zero vector was a
vector space (called the zero vector space).
Because every vector space has a zero vector, the
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 Spring '11
 PaulaTu
 Addition, Multiplication, Scalar, Vector Space

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