## Vector space Definition

Vector space V over Fields F:

V={………..}, the elements of the V is called Vectors

### 1. Addition Composition Law： must obey the axioms of Abelian Group

+ <- V ->
<-
V
->

Note: additive identity 0 here represents zero vector in the vector space, while 0 also can represents additive identity in Field in below, don’t confuse each other

### 2. Scalar multiplication Law:

* <- V ->
<-
F
->

1) closure

∀ c ∈ F, v ∈ V, cv ∈ V

2) associativity

∀ c1,c2 ∈ F, v ∈ V, c1(c2v) = (c1c2)v

3) identity

1 ∈ F, ∀ v ∈ V, 1v = v1=v

4) distributivity

i) ∀ c ∈ F, v1,v2 ∈ V, c(v1+v2) = cv1+cv2

ii) ∀ c1,c2 ∈ F, v ∈ V, (c1+c2)v = c1v+c2v

let c2=0, (c1+0)v = c1v+0v => c1v=c1v+0v, c1v is vector in vector space, both sides apply additive inverse of vector space,

=>(-c1v)+c1v=(-c1v)+c1v+0v => 0 = 0v, note the 0 in the left side represents zero vector because two vectors (-c1v)+c1v add together results in another vector

## V=F

The Field is a vector space over itself

### 2. Scalar multiplication Law:

1) closure

∀ c ∈ F, v ∈ V=F, cv ∈ V=F

2) associativity

∀ c1,c2 ∈ F, v ∈ V=F, c1(c2v) = (c1c2)v

3) identity

1 ∈ F, ∀ v ∈ V=F, 1v = v1=v

4) distributivity

i) ∀ c ∈ F, v1,v2 ∈ V, c(v1+v2) = cv1+cv2 显然满足Field的distributivity

ii) ∀ c1,c2 ∈ F, v ∈ V, (c1+c2)v = c1v+c2v，这个可以先利用Field multiplicative communitive, 然后再用前面的distributivity

(c1+c2)v = v(c1+c2) = vc1+vc2 = c1v+c2v

## V=Fn

Fn, n ∈ N={1,2,3………}

take the arragnements of the little n elements of the field, so the set is going to underlie this vector space Fn is going to be all n tuples of elements from the filed capital F, denote:

 { (x1,x2,……..xn) xi ∈ F }

v=(x1,x2,……..xn) ∈ Fn

v¯=(x1¯,x2¯,……..xn¯) ∈ Fn

v+v¯=(x1+x1¯,x2+x2¯,……..xn+xn¯) ∈ Fn

1) closure, prove by definition of addition

2) ∀ v1,v2,v3 ∈ Fn

(v1+v2)+v3 = v1+(v2+v3)

v1=(x1,x2,……..xn)

v2=(y1,y2,……..yn)

v3=(z1,z2,……..zn)

(v1+v2)+v3 = ( (x1+y1)+z1, (x2+y2)+z2………..,(xn+yn)+zn )

v1+(v2+v3) = ( (x1+(y1+z1), ……………..)

3) identity = (0,0…………0)

v=(x1,x2,……..xn)

-v = (-x1,-x2,……..-xn)

v+(-v) =(x1+(-x1), x2+(-x2),………..) = (0,0…………0)

5) communitative

v=(x1,x2,……..xn) ∈ Fn

v¯=(x1¯,x2¯,……..xn¯) ∈ Fn

v+v¯=(x1+x1¯,x2+x2¯,……..xn+xn¯) ∈ Fn

v¯+v=(x1¯+x1,x2¯+x2,……..xn¯+xn) ∈ Fn

v+v¯=v¯+v

### scalar Multiplication

define scalar multiplication

c ∈ F, v ∈ Fn,

cv=c(x1,x2…….xn) = ( cx1, cx2,……… cxn)

1) closure

∀ c ∈ F, v ∈ V=Fn, cv ∈ V=Fn

 cv= ( cx1, cx2,……… cxn), c, x1,x2…xn∈ F， when we multiple two elements in the field we got another element in the field, so cx1, cx2……….cxn are all elements in the field F, by the definition of Fn: { (x1,x2,……..xn) xi ∈ F },

cv= ( cx1, cx2,……… cxn) ∈ V=Fn

2) associativity

∀ c1,c2 ∈ F, v ∈ V=Fn, c1(c2v) = (c1c2)v

c1(c2v) = c1(c2x1, c2x2,……… c2xn) = (c1(c2x1), c1(c2x2),……… c1(c2xn))

(c1c2)v = ( (c1c2)x1, (c1c2)x2……….. (c1c2)xn)

 同样的by the definition of Fn: { (x1,x2,……..xn) xi ∈ F }, 并且c1,c2 ∈ F，所以

c1(c2x1)= (c1c2)x1。。。。。。。。。。。c1(c2xn)= (c1c2)xn

3) identity

1 ∈ F, ∀ v ∈ V=Fn, 1v = v1=v

4) distributivity

i) ∀ c ∈ F, v1,v2 ∈ V, c(v1+v2) = cv1+cv2 证明方法同上，利用Field的distributivity

ii) ∀ c1,c2 ∈ F, v ∈ V, (c1+c2)v = c1v+c2v，这个可以先利用Field multiplicative communitive, 然后再用前面的distributivity

(c1+c2)v = v(c1+c2) = vc1+vc2 = c1v+c2v

## V=R3

 R3={(x1,x2,x3) xi ∈ R}

the intuitive way of visualizing the vector space is that you can view each one of these as being an arrow basically in three-dimentionaly space, basically you can view each of these vectors as a position vecor where the x coordinate is x1, y coordinate is x2, z coordinate is x3,

what is so wonderful about this is not only does it give us a picture for each vector but also gives us a picture of what it means to add two vectors together and scalar multiply vector