Linear Algebra: Prove that if $T$ is a linear map from vector space $V$ to vector space $W$, then the range of $T$ is a subspace of $W$ So, I am really having a hard time with linear algebra this semester, particularly when it comes to proofs. I have never done proofs before this class, so I was wondering if I could get some guidance on what I have so far.
I basically start off by saying: 
Suppose we have two arbitrary vectors $a_1$ and $b_1$ within vector space V. 
Where $T(a) = w_1$ and $T(b) = w_2$.
To satisfy linearity, $T(a + b) = T(a) + T(b) = w_1 + w_2$ and $T(ca) = cT(a) = w_1$ & $T(cb) = cT(b) = w_2$ where $c4$ is an arbitrary scalar in $R$.
Because $T(a) = w_1$ and $T(b) = w_2$, $w_1$ and $w_2$ are both in the range of $T$. 
We must now satisfy the subspace test for $w_1$ and $w_2$.
It is at this point where I am a bit stuck. I am not really sure how to prove that $w_1$ and $w_2$ are actually a subspace without having actual numbers to work with. I don't want to just say that they are via the subspace test, because I feel like I have basically proven nothing. Up until this point I basically feel as if I've proven nothing.
Is there any guidance someone could give?
Thanks.

You just demonstrated that the range is closed under addition and scalar multiplication; you are done.
