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Vector spaces and vector subspaces.
In this tutorial,
denotes any field (generally
or
).
Definition 5.1.1
On the set
V we define two operations, one named
addition and denoted by +, the other one named
multiplication by scalars and denoted
(sometimes we can omit the dot).
The triple
is called
a vector space (actually a vector space over
)
if the
following properties are fulfilled:
 1.
 Closure:
 2.
 Associativity:
 3.
 Neutral element: There exists an element, called
the zero vector and denoted by
,
such that
 4.
 Additive inverse: for any vector
,
there exists a vector denoted
such that
 5.
 Commutativity:
 6.
 Closure:
 7.

,
 8.

,
 9.

 10.

,
Definition 5.1.3
Let W be a subset of the vector space V. This subset is called a
vector subspace of V if it is a vector
space, when the addition and multiplication by scalars are the restriction to W of
those of V.
Proposition 5.1.4
Let
V be a vector space over
,
where
is any field and
let
W be a non empty subset of
V. Then
W is a (vector) subspace of
V if,
and only if, it fulfills the following requirements:
 1.
 W is closed under addition.
 2.
 W is closed under multiplication by scalars.
The other properties are trivially fulfilled, excepted the belonging to W of
the zero vector, which is a consequence of the second requirement.
Figure 1:
A linear combination of two vectors

A linear combination of the vectors
is a vector of the form
where only a finite number of the coefficients is not zero.
Proposition 5.1.5
The subset W of V is a (vector) subspace of V if, and only if, it is closed under
linear combinations.
The ``if'' part is obtained by setting
for the closure under
addition, and
for the closure under multiplication
by scalars. The ``only if'' part is quite trivial.
Proposition 5.1.7
Let
U and
V be two vector subspaces of the same vector space
W. Then
is a subspace of
W.
Note that the corresponding result for the union of two subspaces is false: e.g.
take
We have
and
,
but
(1,0)+(0,1)=(1,1) and
.
Next: Linear dependence and independence.
Up: Vector spaces.
Previous: Vector spaces.
Noah DanaPicard
20010226
+00wE