Working through a homework problem from my linear algebra course. We're using the Gareth Williams Linear Algebra with Application - 7th Edition. The question comes from section 5.2: Matrix Representation of Linear Transformations
Consider the linear transformation $T:R^2 \rightarrow R^2$ defined by $T(x, y)=(2x, x+y)$. Find the matrix $T$ with respect to the basis $\{u_1, u_2\}$ of $R^2$ where: $u_1=(1, 2)$ and $u_2=(0, -1)$
Here is my process thus far, I may be over thinking things a bit.
First I evaluate the transformation on the standard basis $\{e_1, e_2\}$. $T(e_1)=(2, 1)$ and $T(e_2)=(0, 1)$
Then: $c_1u_1 + c_2u_2 = (2, 1)$ and $c_1u_1 + c_2u_2 = (0, 1)$
Solving for $c_1$ and $c_2$ I get the coordinate vectors $[T(u_1)]_B = (2, 3)$ and $[T(u_2)]_B = (0, -1)$
I should then get the transition matrix $T=[[T(u_1)]_B, [T(u_2)]_B]$ (I'm not sure how create matrices but it should be the $2 \times 2$ matrix with column vectors $(2,3)$ and $(0, -1)$)
I'm then to find the image of $(-1, 3)$ which I believe I would just left multiply by $T$. My issue is that both my $T$ and my image are nowhere close to what the solution in the back of the text says they should be. I've led myself astray somewhere here and I'm lost as to where. Any help is appreciated!