Wednesday February 6
−
Lecture 15:
More on abstract vectors spaces and subspaces
(Refers to section 4.1)
Expectations:
1.
Determine if a given set is a vector space or not.
2.
Recognize linear transformations from one vector space to another.
15.1
Definition –Let {
v
1
,
v
2
, …,
v
m
} be a finite set of vectors in a vector space
V
.
If every
vector
v
in
V
is a linear combination of
v
1
,
v
2
, …,
v
m
then we say that spans
i
and write
span{
v
1
,
v
2
, …,
v
m
}.
15.1.1
Remark – As for
R
n
if
W
= span{
u
1
,
u
2
, …,
u
m
} inside a vector space V the
W
is a subspace of
V
.
15.1.2
Example – We show in class that the set {1,
x
,
x
2
, …,
x
n
} spans
P
n
, i.e.,
P
n
=
span{1,
x
,
x
2
, …,
x
n
}.
15.2
Definition – A function
T
which maps vectors in a vector space
V
1
to vectors of
another vector space
V
2
is called a linear map if it satisfies the condition
T
(
α
v
+
β
w
) =
α
T
(
v
) +
β
T
(
w
).
•
Note that the vectors spaces need not be similar in nature. That is
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Winter '08
 All
 Linear Algebra, Algebra, Transformations, Vectors, Vector Space, vector space V1

Click to edit the document details