Find a linear transformation defined by $T(0,1,2) = (3,1,2)$ and $T(1,1,1) = (2,2,2)$.

Question

The transformation here (as per my calculations) will be $T\colon U \rightarrow V$ such that $T(x,y,z) = (y+z, 3y-z, 2y)$ where $z=-x+2y$.

Now what should $U$ be $\Bbb R^3$ or a subset of $\Bbb R^3$?

Answer 1

If you want a linear transformation $T$ such that $T(u_1)=v_1$ and $T(u_2)=v_2$ where $u_1$ and $u_2$ are linearly independent, then we want $T(\alpha u_1+\beta u_2)=\alpha T(u_1)+\beta T(u_2)$ for scalars $\alpha,\beta$, and then $T$ is defined on the subspace $\operatorname{span}(u_1,u_2)$. Also, the range will be $\operatorname{span}(v_1,v_2)$.