Let $T : R^2 → R^3$ be the linear transformation given by a rule Find the matrix $A$ that represents $T$ relative to ordered bases

Question

Let $T : R^2 → R^3$ be the linear transformation given by the rule $T(x, y) = (2x−y, 3y, 2y−3x)$ Find the matrix $A$ that represents $T$ relative to the ordered bases $B = {(1, 1),(0, 1)}$ and $B ′ = {(1, 0, 0),(1, −1, 0),(1, 1, 1)}$ of $R^2$ and $R^3$, respectively. Can anyone help me out here, Don't really understand what to do. Any hint would be much appreciated, Thanks!

Answer 1

You have to decompose $T(1,1)$ and $T(0,1)$ in the basis $B'$. For example $$T(1,1)=(1,3,-1)=6\cdot(1,0,0)+(-4) \cdot(1,-1,0)+(-1) \cdot (1,1,1)$$ so the first column is: $$ \begin{pmatrix} 6 & \bullet\\ -4 & \bullet \\ -1 & \bullet \end{pmatrix} $$