# Help with simple linear transformation

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.

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