Concept of a basis for a vector space

[mrroboto]

concept of a "basis" for a vector space

I do not understand the concept of a "basis" for a vector space.

Here's an example from my practice final exam:

Suppose U and V are subspaces of the real vector space W and {u1} is a basis for U and {v1} is a basis for V. If U intersection V = {0} show that {u1, v1} is a linearly independent set it W.

I probably need additional help with this example, but if someone could explain a "basis" to me in terms of this example I would greatly appreciate it.

Thanks.

 

[EnumaElish]

Any vector in U can be expressed as a linear transformation of u1 (e.g., k*u1). Any vector in V can be expressed as a linear transformation of v1 (e.g., k*v1).

 

[mrroboto]

Ok, thanks!