Espaços vetoriais

Vector spaces

Definition 1. A real vector space is a set of elements together with two operations satisfying the following properties: If and the operation a). are any elements of ) for any ( ( in , , then is in (i.e., closed under

; for any such that in such that , ; for any ( ; ;

b).

c). There is an element d). For each in

, there is an element

If is any element of closed under the operation e). f). ( g). h). 1 ( ( (

and )

is real number, then

is in (i.e.,

for all real numbers

and all

and

in

The elements of The operation multiplication. The vector The vector

are called vectors; the real numbers are called scalars. is called vector addition; the operation is called scalar

in property c) is called a zero vector. in property d) is called a negative of

Example 1. Consider the set ( numbers. If put ( ( , ) and { ( and ( }, where is the set real ) then we

Then the set is vector space under the operations of addition and scalar multiplication of - vectors.

Example 2. Consider the set of all ordered triples of real numbers of the form ( and define the operations and by ( ( ( ( ( .

is vector space, since it satisfies all properties of Definition 1.

Example 3. (

Consider the set

and define the operations ( ( .

and

by

Properties a), b), c), d) end e) of definition hold. Here the negative of the vector ( is the vector ( For example, for verify property e) proceed as follows. First, ( ( Also ( [( ( ( ( ( ( ] ( (

(

is zero and .

(

We show that property f). fails to hold. Thus ( ( ( ) (( (( ( (( the property f). is not valid. ). (

On the other hand, ( ( As (( Therefore, Example 3.

is not vector space. Consider the set

{[

]

} of all

matrices under the usual is vector space.

operations of matrix addition and scalar multiplication. Then Example 4. Let on the interval [ [ ] ]. If

be the set of all real valued functions the are defined [ ] we define by ( ( ( by ( ( .