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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.
,

Example 5.1.2
1.
The field is a vector space over .
2.
The space is a vector space over .
3.
The set is not a vector space over , as the first property does not hold.

Take and in , then .

4.
The set of all continuous functions on the interval [0,1] is a vector space over . For a proof, see your favorite Calculus course.

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.
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.

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.

Example 5.1.6
1.
The set of all odd functions from to itself and the set of all even functions from to itself are vector subspaces of the space , which elements are all the functions from to itself.
2.
The set

is not a vector subspace of .

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 Dana-Picard
2001-02-26
+00|wE