# Basis of a Vector Space

A subset of a vector space is said to be a basis of , if

(i) consists of linearly independent vector, and

(ii) generates , i.e. i.e. each vector in is a linear combination of finite number of elements of .

For example the set is a basis of the vector space over the field of real numbers.

__Dimension__**:**

The dimension of a vector space is the number of elements in a basis of .

__Example__**:
**Show that the set forms a basis for .

__Solution__**:**

For , then

Hence the given set is linearly independence.

Now let

So that

Thus, the unit vector is a linear combination of the vectors of the given set, i.e.

Since is generated by the unit vectors, , we see therefore that every elements of is a linear combination of the given set . Hence the vectors of this set form a basis of .