Vector spaces and sub spaces: Part-ii

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