Find linear transformation of a given matrix of linear transformation

Question

I have two basis.

The first is a $R2$ basis ${(1,0) , (0,2)}$. Lets call it basis of $U$.

The second is a $R3$ basis ${(1,0,-1), (0,1,2), (1,2,0)}$ Lets call it basis of $V$.

Is given a matrix of a linear transformation: $[T]$ $/ U -> V$

$[T]$ =

|1   0|
|1   1|
|0  -1|

I wanna know what is the linear trasformation $T$!

Thank you!

Answer 1

Let's denote $U_c$ and $V_c$ the canonical basis of $\Bbb R^2$ and $\Bbb R^3$ respectively and let $P$ and $Q$ the change matrices basis from $U_c$ to $U$ and from $V_c$ to $V$ respectively then $$[T]_{U_c\to V_c}= Q[T]_{U\to V}P^{-1}$$ hence we find $$T(x,y)=\left(x-\frac12y,x-\frac12y,x+y\right)$$