A vectorspace with 1 basis vectors is a straight line.

A vectorspace with 2 basis vectors is a plane.

A vectorspace with 3 basis vectors is a space.

So does such a thing as a 4 basis vectors exist in a vectorspace?

If a 4th vector existed, then this 4th vector could always be written as a linear combination of the 3 other vectors no matter what, and the vectors of a basis always have to be linearly independent, thus a 4th vector would make the vector set linearly dependent (since they can be written as a linear combination)... so my guess is that a basis can never be more than 3 vectors?