- #1
snipez90
- 1,101
- 5
Homework Statement
Let V and W be vector spaces over F, and let T: V -> W be a surjective (onto) linear map. Suppose that {v1, ..., v_m, u1, ... , u_n} is a basis for V such that ker(T) = span({u1, ... , u_n}). Show that {T(v1), ... , T(v_m)} is a basis for W.
Homework Equations
Basic properties of linear maps. Linear independence.
The Attempt at a Solution
I have already proven that {T(v1), ... , T(v_m)} spans W, which I thought would be harder than showing linear independence. But here is where I am confused. We have to show that {T(v1), ... , T(v_m)} is linearly independent.
Suppose (a_1)T(v1) + ... + (a_m)T(v_m) = 0 for (a_1), ... (a_m) in F. Then
T( (a_1)(v1) + ... + (a_m)(v_m) ) = 0 since T is linear. Now if we also suppose that (a_1)(v1) + ... + (a_m)(v_m) = 0, then clearly (a_1) = (a_2) = ... = (a_m) = 0 since the set {v_1, ... , v_m} is linearly independent.
But I think I'm confused when (a_1)(v1) + ... + (a_m)(v_m) =/= 0 (which is certainly possible right?). However, I have an idea and I think that in this case, we still get (a_1) = ... = (a_m) = 0?