I am having problems in understanding the following exercise: $$F: \mathbb{R}\to \mathbb{R}^2; \qquad F(x) = (2x, 3y)$$ I have to say if it's linear.
I am having troubles in understanding where does the $y$ come from. Anyway, Shall I prove by linearity that
$$F(x_1 + x_2) = F(x_1) + F(x_2)$$ hence
$$F(x_1 + x_2) = (2(x_1 + x_2), 3y) = (2x_1 + 2x_2, 3y)$$
I think by sum of vectors that I cannot say that the previous is linear since it should be
$$(2x_1, 3y) + (2x_2, 3y)$$
Which is false.
Am I right?
You should understand $y$ as being a fixed constant i.e. its value does not change when the value of the variable $x$ changes.
To check linearity let $x,x' \in \mathbb R$, then:
$$ F(x + x') = F(x)+F(x') \iff (2x+2x',3y) = (2x,3y) + (2x',3y)$$
$$ \iff 3y = 6y$$
$$ \iff y = 0.$$
So the property $F(x+x') = F(x)+F(x')$ is only verified if $y = 0$.
If $y = 0$ then
$$ F(\lambda x) = \iff (2\lambda x,0) = \lambda (2x,0) = \lambda F(x).$$
Hence the map is not linear when $y \neq 0$ and linear when $y = 0$.