Vector Space
Set of elements called vectors, with 2 operations with certain properties:
Usually we will note a vector in boldfaced letters or
with an arrow above it
1 ) Addition of vectors : commutative, associative, with an identity element
and each element must have inverses
- u + v = v + u
- u + ( v + w ) = (u + v
) + w
- there is 0 such as for any vector v, 0 + v =
v
- for every vector v, there is another vector w such as v
+ w = 0 . w is written " -v "
2 ) Scalar Multiplication : associative and distributive
- (ab)v = a (bv)
- 1v = v
- (a + b)v = av + bv
- a(v + w) =av + aw
Given those two operations, we may define a "linear combination"
of a set of vector v1, ... vn : any vector of the form
a1v1 + a2v2 + ... anvn
Example of Vector Space : Rn (the set of all ordered n-tuples of real number)
is a vector space
Sub example : R3 :
/ r1 / 1 / 4 / 5
| r2 ...... ex : | 3 + | 5 = |12
\ r3 \ 6 \ 2 \ 8