1. ## Images, Ranges question...

Last question for the night!

Let V and W be vector spaces and T: V --> W be linear.

a) Prove that T is 1-1 if and only if T carries linearly independent subsets of V onto linearly independent subsets of W.

b) Suppose that T is 1-1 and that S is a subset of V. Prove that S is linearly independent if and only if T(S) is linearly independent.

c) Suppose B {v1, v2,...,vn} is a basis for V and T is 1-1 and onto. Prove that T(B) = {T(v1), T(v2),...,T(vn)} is a basis for W

2. Still working on it! Any help would be appreciated

3. suppose T is 1-1 and S is a linearly independent subset of V. If T(S) isn't linearly independent then you have
$0=\sum_{i=1}^n c_i T(v_i) = T(\sum_{i=1}^n c_i v_i)$
where $v_i \in S, c_i \neq 0$ and because T(0)=0 and T is 1-1 you have $0=\sum_{i=1}^n c_i v_i$
so S is not linearly independent and you get a contradiction. meaning T is 1-1 then T(S) linearly independent if S is linearly independent.

if T carries linearly independent subset to linearly independent subsets, then for every nonzero vector $v \in V, \left \{ v \right \}$ is independent and then {T(v)} is independent so $T(v) \neq 0$, and T is 1-1 because it's kernel is only the zero vector

the proof for (b) is similar to (a)

for (c) you already know that T(B) is independent. try to show that is also spans W using the fact that T is onto.