[Math] Groups, Vector Spaces and Vector Subspaces

**My Personal opinion

In linear algebra, finding solution x from Ax = 0 is very important.

And possible solution set x forms a vector subspace which is included in R^n vector space.

Before defining the solution space, What we will do is to look into what group, vector space, vector subspaces are.

Simply, group gurantees that the results from the operation still lives in the same group(has more specific properties to satisfy)

Vector space is group of vectors. (but also satisfies certain axioms)

1. Groups

Group is a set of elements and an operation defined on these elements that keeps some structure of the set intact.

Consider a set G and an operation op, G op G -> G defined on G. Then (G, op) is called a group if the following hold.

1. Closure of G under op : all x, y ∈ G : x op y ∈ G
2. Associativity : all x, y, z ∈ G : (x op y) op z = x op (y op z)
3. Neutral element : any e ∈ G, all x ∈ G : x op e = x and x op e = x
4. Inverse element : all x ∈ G, any y ∈ G : x op y = e and y op x = e, where e is neutral element.

Then (G, op) is a group.

• If additionally commutative : all x, y ∈ G : x op y = y op x, then G = (G,op) is an Abelian group

(Z, +), integer set Z
1. ok
2. ok
3. ok
4. ok
5. ok
-> Abelian group

(N_0, +) Natural number including 0, integers not smaller than 0
1. ok
2. ok
3. ok, 0
4. fail
-> not a group

(Z, *)
1. ok
2. ok
3. ok, 1
4. fail, 1/x is not always integer
-> not a group

(R, *), real number
1. ok
2. ok
3. ok, 1
4. fail, 1/x doesn't always exist when x is 0
is not a group

(R\{0}, *)
1. ok
2. ok
3. ok, 1
4. ok, 1/x
5. ok
-> Abelian group

(R^n, +) n ∈ N, e.g (x1, ..., xn)
1. ok
2. ok
3. ok, (0, ..., 0)
4. ok, (-x1, ..., -xn)
5. ok
-> Abelian group

(Z^n, +) n ∈ N
1. ok
2. ok
3. ok, (0, ..., 0)
4. ok, (-x1, ..., -xn)
5. ok
-> Abelian group

(R^m*n, +) m,n ∈ N
1. ok
2. ok
3. ok, (0, ..., 0)
4. ok, (-x11, ..., -xmn)
5. ok
-> Abelian group

(R^m*n, *) m,n ∈ N
1. ok
2. ok, matrix multiplication, (AB)C = A(BC)
3. ok, I, identity matrix
4. ok if A^-1 exists
-> we call it general linear group

2. Vector Spaces

Vector space is a structured space in which vector lives.

A real-valued vector space (V, +, *) is a set V with two operations

+ : V + V -> V

* : R * V -> V

where

• 1. (V, +) is Abelian group
• 2. Distributivity:
  • all λ ∈ R, x,y ∈ V : λ(x+y) = λx + λy
  • all λ, ψ ∈ R, x ∈ V : (λ + ψ) * x = λx + ψx 
• 3. Associativity (outer operation) : all λ, ψ ∈ R, x ∈ V : λ(ψx) = (λψ)x 
• 4. Neutral element w.r.t the outer operation: all x ∈ V : 1*x = x

V = R^n, n ∈ N is a vector space with operations defined follows
Addition : x + y (x, y ∈ R^n)
Muliplication by scalars : λx (for all λ ∈ R, x ∈ R^n)

V = R^m*n, m,n ∈ N is a vector space with
Addition: A + B (A, B ∈ V)
Multiplication by scalar: λx (for all λ ∈ R, x ∈ R^n)

3. Vector Subspaces, Linear Subspaces

A subset U of vector space V is a vector subspace or linear subspace.

To determine whether (U, +, *) is a subspace of V we need to show

• U is not null and 0 ∈ U
• Closure of U
  • w.r.t outer operation : all λ ∈ R, all x ∈ U : λx  ∈ U
  • w.r.t outer operation : all x, y ∈ U, x + y ∈ U

Only D is a subspace of R^2

A,C violates closure property, and B doesn't contain 0

 

The solution set of a homogeneous system of linear equations Ax=0 with n unknowns x = [x1, ..., xn].T is a subspace of R^n

The solution set of Ax = 0 forms a vector space because

• It always includes the zero vector (0, 0, ..., 0) as a solution.
• If x1 and x2 are solutions, then their sum (x1 + x2) is also a solution.
• If x is a solution and c is any scalar, then cx is also a solution.
• R^n is a vector space