**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 V

_{2 }and its rank = 2.

ii) (1, 0, 0); (0, 1, 0); (0, 0, 1) is a basis of V

_{3 }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

_{1 }& S

_{2 }are such that every vector of S

_{1 }is also a vector of S

_{2 }. Then S

_{2 }is called Super set and S

_{1 }is its Sub-set.

Example: (1, 0, 0) ; (0, 1, 0); (0, 0, 1) → S

_{1 }→ Sub-set of S

_{2 }

(1, 0, 0) ; (0, 1, 0); (0, 0, 1); (1, 2, 3) → S

_{2 }→ Sub-set of

_{ }S

_{1 }

Vector Spaces are Supper set and Sub-spaces are sub-set.

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