Vector spaces and sub spaces: Part-i

Vector space is set of vectors which are closed under addition and scalar multiplication.
A set of vectors is said to be 'closed under addition' if  the sum of any two vectors is also a vector of the same set.
A set of vectors is said to be 'closed under multiplication' if the scalar multiplication of any vector is also a vector of the same set.
If A & B are two vectors from the set so does (A+B) and also λA is also another vector of same set. 
Example of vector spaces:
V₂ → {x, y} represents a vector space of order 2 for different values of x & y.
V₃ → {x, y, z}
V_n → {x₁, x₂, ---, x_n}
Totality of vector obtained by linear combination of  given set of vectors is called subspaces of the given vector set.

Generating or Spanning set:
Generating set is set of vector whose linear combination generate or span a vector space.
 Basis of  a vector space is a set of linearly independent vectors that generate on entire vector space.
For example,
(1,0,0); (0,1,0); (0,0,1) is a basis of vector space V₃ .
Let (x,y,z) be a vector from  V₃ . Clearly-
x (1,0,0) + y(0,1,0) + z(0,0,1) = (x,y,z)