prove: transformation T(x, y, z)=(2x, y)

Question

I am familiar with the process of proving linearity but I haven't seen an example of $\mathbb{R^3}>\mathbb{R^2}$ with $\mathrm{T}(x, y, z)=(2x, y)$.

Should I go ahead, start with this $> (2(x1+x2), (y1+y2))$ and then continue? Or am I neglecting the $z$ coordinate?

Answer 1

It's exactly the same as the case of both the domain and the image being of the same dimension.

Let $\lambda(x_1+x_2, y_1 + y_2, z_1 + z_2) \in \mathbb D_T, \lambda \in \mathbb R$. Then, plug this in your transformation and use the vector properties to prove that it's just a linear combination of each while also proving the scalar multiplication property.