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Math 20F Linear Algebra
Lecture 9: 4.1 Vector Spaces.
Recall that if
x
,
y
∈
R
n
are vectors in Euclidean space we deﬁned the
addition
x
+
y
∈
R
n
and
scalar multiplication
λ
x
∈
R
n
. In dimensions 2 and 3 we can
deﬁne these notions geometrically
The addition and scalar multiplication satisfy certain properties listed below, but
these properties show up in many diﬀerent contexts so rather than studying each
situation individually we will study them all at once.
A set
V
with two operations, addition and multiplication by scalars, deﬁned on it is
called a
vector space
if the following properties hold for any
u
,
v
,
w
∈
V α,β
∈
R
:
1. If
u
,
v
∈
V
then
u
+
v
∈
V
. (closure under addition)
2.
u
+
v
=
v
+
u
(commutative)
3. (
u
+
v
) +
w
) =
u
+ (
v
+
w
) (associative)
4. There is an element
0
∈
V
such that
u
+
0
=
u
all
u
∈
V
(additive unit)
5. For each
u
∈
V
there is

u
∈
V
such that
u
+ (

u
) =
0
(additive inverse)
6. If
u
∈
V
and
α
is a scalar then
α
u
∈
V
. (closure under scalar multiplication)
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