Question

can somebody explain to me the basis & dimension of a vector space, with example?

Answer 1

Hello @akifnadeem ,
Instead of giving you a formal difintion which you can easily find in wikipedia ( http://en.wikipedia.org/wiki/Dimension_(vector_space))

For the intuitive explantion:
Any vector space contain a basis. The basis is a linearly independent subset of V which spans V.
For example if you look at the vector space \[R^{2}\] then the standard basis is {(1,0 , (0,1)} because these are linearly independent and also span V.

The number of vectors in the basis is called dimension.

Notice that the subset {(1,0) , (0,1), (1,1)} also spans V but isn't linearly independant and therefore isn't a basis.

Also the subset {(1,0)} is linearly independant but doesn't span V so it's a basis.

That is, once you define your basis, the dimension is just the number in the subset you define.

Hope it helps you.