**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:**

V

_{2}→ {x, y} represents a vector space of order 2 for different values of x & y.

V

_{3 }→ {x, y, z}

V

_{n}→ {x

_{1}, x

_{2}, ---, 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 V

_{3 }.

Let (x,y,z) be a vector from V

_{3 }. Clearly-

x (1,0,0) + y(0,1,0) + z(0,0,1) = (x,y,z)

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