As the title states I have:
Let T : V → W be a linear transformation.
Prove that if $v_1$,..., $v_n$ is a generating set of V, then $T(v_1),...,T(v_n)$ generate
the image $im(T)$.
I think the problem wants me to prove this:
Suppose that any $v∈V$ can be written as $v=v_1+,...v_2$(definition of a generating set) then any $T(v)∈Im(T)$ can be written as $T(v)=T(v_1)+...T(v_2)$
I'm unsure of where to start in proving these conditions. Should I be using the definition of a linear transformation $T(v_1+v_2)=T(v_1)+T(v_2)$ and $T(cv)=cT(v)$ to try and prove this or is that the wrong direction?
You basically got it. You noted that for $w \in \mathrm{Im}(T)$, you have a $v \in V$ such that $T(v) = w$ by definition of $\mathrm{Im}(T)$ and you noted that by linearity of $T$, you have $w = \alpha_1 \cdot T(v_1) + \dots + \alpha_n \cdot T(v_n$). Since $w$ is any vector of $\mathrm{Im}(T)$, that means that $T(v_1), \dots T(v_n)$ generates $\mathrm{Im}(T)$.
Just make sure to mention the terminology in italics. And don't forget the scaling factors $\alpha_i$ in a linear combination.