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.
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.
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 .