Help with simple linear transformation

Question

can you please help me with this simple question?

I want to know if this is linear or not

$f: \Bbb R^2\to\Bbb R^3$

$f(0,0)=(1,0,0)$

$f(0,1)=(0,0,0)$

If you could explain why it is or it isn’t linear, I’d be really grateful.

All the best.

Answer 1

Every linear transformation fix the origin, i.e. $T(0)=0$

Answer 2

To expand on the answer of rowcol, the definition of a linear operator is that it satifies $f(\alpha x + \beta y) = \alpha f(X) + \beta f(y)$. Note that if you set $\alpha = 0$ and pick any $x \in \mathbb{R}^2$, you get $$ f(0 x) = f(0) = (1, 0, 0) \neq 0 f(x). $$ Therefore, $f$ is not linear.