Vector spaces and sub spaces: Part-ii

Rank or Dimension  of a Vector Space or Sub-space:
Rank or dimension of a vector space or sub-space is the number of independent vectors in the basis of that space or sub-space. 
For example,
i) (1,0); (o,1) is a basis of V₂ and its rank = 2.
ii) (1, 0, 0); (0, 1, 0); (0, 0, 1) is a basis of V₃  and its rank = 3.

Important Note:
To be a vector space there must have at least three independent vectors. But to be vector sub-space, one or two independent vectors are enough.

Orthogonal and Orthonormal Basis of Vector Space and Sub-space:
Orthogonal basis is a basis of vector space whose all vectors are pairwise orthogonal.
Example: All unit vectors.
(1, 0, 0) ; (0, 1, 0); (0, 0, 1)

Super set and Sub-set of Vectors:
Two sets
 of vectors S₁  & S₂  are such that every vector of S₁  is also a vector of S₂ . Then S₂  is called Super set and S₁ is its Sub-set.
Example:  (1, 0, 0) ; (0, 1, 0); (0, 0, 1) → S₁  → Sub-set of S₂ 
  (1, 0, 0) ; (0, 1, 0); (0, 0, 1); (1, 2, 3) → S₂  → Sub-set of  S₁
Vector Spaces are Supper set and Sub-spaces are sub-set.