range T = span{(1,2,3),(1,1,1),(0,2,4)} and
null space of linear map T = { x belongs to R^4 | x[1]+x[2]+x[3]+x[4]=x[1]+2*x[2]+x[3]-x[4] = 0}.
**x[1]: [1] is a subscript, meaning the first term in the list x.
2. Let T: V->W be linear and surjective. Let B be a basis for W, choose for each basis vector {w belongs to B} a vector {v = f(w) belongs to V} such that Tv=w.
Prove that the set A = f(B) = {f(w) | w belongs to B} (which contains as many elements as the set B) is linearly independent.
What conclusion follows regarding the dimensions of V and W?
更新1:
For (2), after proving v's are linearly independent, how can we be sure that v is a basis of V? I am not sure if it implies that v's span V as the set A seems only a subset of V.
