# Finding the system of linear equations defining a linear transformation

Heres the textbook solution: The goal is to find a system of linear equations that defines some linear transformation. In this case TU.
Given:

\begin{align}T:y_1 &= −3x_1 + x_2\\ y_2 &= x_1 − x_2 \end{align}

\begin{align}U: y_1 &= x_1 + x_2 \\ y_2 &= x_1 \end{align}

Trying to solve for $TU$
Attempt:
\begin{align}T(c_1,c_2)&=(-3c_1 + c_2, c_1 -c_1) \\ U(b_1,b_2)&=(b_1+b_2,b_1) \\ T[-3(b_1+b_2)+b_2, b_1) &= -3b_1-2b_2,b_1\\ y_1&=-3x_1-2x_2 \\ y_2&=x_1 \end{align}

However the solutions manual has the answer as $y_1=-2x_1 - 3x_2$ and $y_2=x_2$

Wheres the error?

$$U(b_1, b_2) = (b_1 + b_2, b_1)$$
$$T(c_1, c_2)= (-3c_1+c_2, c_1-c_\color{red}2)$$
$$T(U(b_1, b_2))=T(\color{red}{b_1+b_2}, b_1)=(-3(b_1+b_2)+b_1, (b_1+b_2)-b_1)=(-2b_1-3b_2, b_2)$$