Why does a basis for $\mathbb{R}^n$ have to be linearly independent? I know by definition a basis for $\mathbb{R}^n$ needs to be linearly Independent. However, is there a reason for why it has to be? 
 A: We choose definitions but there are reasons why it is a useful choice.
When a set of vectors does not span $\mathbb{R^n}$ (or a certain subspace), you can add a vector to this set that isn't in the span to enlarge the spanned space. You can continue this proces until the desired space is spanned by this set of vectors.
In a set of vectors which is linearly dependent, you can always omit a vector without changing the subspace spanned by those vectors. You can continue discarding these obsolete vectors until the set becomes linearly independent.
You could look at a basis as a set of vectors where two interesting properties come together:


*

*you have enough vectors to span the entire space;

*there are no (unnecessary) vectors 'in excess'.


In summary, a basis is thus a minimal, spanning set.
This definition of a basis guarantees that every vector of the space $\mathbb{R^n}$ can be written as a unique linear combination of the vectors of the basis. This leads to the meaningful introduction of coordinates (with respect to this basis), precisely the (ordered) coefficients of this unique linear combination.
