Find the linear transformation matrix

[discy]

Homework Statement

Suppose that a linear transformation maps a point (2,3) to (0,1) and maps a point (9,7) to (1,0). Find the matrix for this linear transformation.

2. Solution (answersheet)
Observe that the two point that are the result of the mapping are the two base vectors.
If our information would be that (0,1) were mapped to (2,3) and that (1,0) were mapped to (9,7), then the matrix would be easy to write down.
|9, 2|
|7, 3|
But we are in the opposite direction! so the answer is the inverted matrix.

3. My question
How would this question be solved if the points wouldn't map to the base vectors? I.o.w. what are the common steps to solve this, that are skipped here? If possible. The only way I know is with homogeneous coordinates, but I couldn't get it to work with this exercise.

 

[lippyka]

Thanks in advance. A:To answer your question, assume that the linear transformation maps the two points $(x_1,y_1)$ to $(a_1,b_1)$ and $(x_2,y_2)$ to $(a_2,b_2)$. Then the matrix for this transformation is$$A=\begin{bmatrix} a_1 & a_2 \\b_1 & b_2\end{bmatrix}\begin{bmatrix} x_1 & x_2 \\y_1 & y_2\end{bmatrix}^{-1}$$where the inverse of the second matrix is given by$$\begin{bmatrix} x_1 & x_2 \\y_1 & y_2\end{bmatrix}^{-1}=\frac{1}{x_1y_2-x_2y_1}\begin{bmatrix} y_2 & -x_2 \\-y_1 & x_1\end{bmatrix}.$$In your example, the two points are $(2,3)$ and $(9,7)$, which map to $(0,1)$ and $(1,0)$. Therefore the matrix of the linear transformation is$$A=\begin{bmatrix} 0 & 1 \\1 & 0\end{bmatrix}\begin{bmatrix} 2 & 9 \\3 & 7\end{bmatrix}^{-1}=\frac{1}{15}\begin{bmatrix} 7 & -9 \\-3 & 2\end{bmatrix}.$$