How can I verify that the following function is linear?
L : $R^2 \Rightarrow R^3$ such that $L(v_1,v_2) = (v_2,4v_1+v_2,0)$
I know that the theorem states that:
for all $v \in R^m$ and all $\alpha \in R^m$, we have $L(\alpha v) = \alpha L(v)$.
for all $v,w \in R^m$, we have $L(v + w) = L(v) + L(w).$
But I need some help starting with the arithmetic of the proof, how do I do this when they are in different dimensions?