# Linear Transformation - How can I verify/prove that the following function is linear?

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:

1. for all $$v \in R^m$$ and all $$\alpha \in R^m$$, we have $$L(\alpha v) = \alpha L(v)$$.

2. 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?

• $L(\alpha (v_1,v_2))=L(\alpha v_1,\alpha v_2)$. Compare with $\alpha (L(v_1,v_2))=\alpha (v_2,4v_1+v_2,0)$. – Mauro ALLEGRANZA Sep 11 at 14:28
• The same for 2. – Mauro ALLEGRANZA Sep 11 at 14:33
• Quibble: "how can I show the following linear transformation is linear" is silly. If it is a linear transformation, then by definition it is linear. You mean, how can you show the given function is in fact a linear transformation. – Arturo Magidin Sep 11 at 14:37
• That’s not a “theorem.” That’s the definition of linearity. – amd Sep 11 at 16:57

$$L(\alpha v) = (\alpha v_2, 4 \alpha v_1+\alpha v_2, 0)$$
$$\alpha L(v) = (\alpha v_2, 4 \alpha v_1+\alpha v_2, 0)$$
Are these the same for all $$v$$?
• If i try to solve for $L(\alpha v)$ using the individual componets of $v_1$ and $v_2$, what do you suggest I name the components of $v_1$? For example if I was using $x$ and $y$, I would use ($x_1,x_2$) and ($y_1,y_2$).. – atn Sep 11 at 15:10