About the basis of a quotient space

[sanctifier]

Notations:
V denotes a vector space
S denotes a subspace of V
V/S denotes a quotient space
V\S denotes the complement of S in V

Question:
If {s₁, ... , s_k} is a basis for S, how to find a basis for V/S?

I realize that the basis of V\S may determine the basis of V/S, but I don't know how to formulate it. For example, let R² be the Cartesian plane V^, with basis {(1,0),(0,1)}, the diagonal is a subspace S^, whose basis is {(1,1)}, then how to formulate the basis of V^,\S^,?

Thanks for any help!

 

[quasar987]

It is indeed a good guess that a basis for V\S determines one for V/S. Namely, that if p:V-->V/S is the quotient map (aka the projection map) and if (v_1,...,v_{n-m}) is a basis for V\S, then (p(v_1),...,p(v_{n-m})) is a basis for V/S. To prove it though, you still need to show that this set of vectors is linearly independent and generated V\S.

 

[sanctifier]

Thanks!