The key idea of this problem is to realize that the plane contains linearly independent vectors; For example, .
Therefore, if is a linear transformation that maps into , in fact sends all of to . In particular, we have
for some constants
On the other hand, we also need to ensure that at least one point of is not mapped to the zero vector: otherwise the image of under would simply be the zero vector instead of all of . In other words, at least one of , , and is non-zero.
Therefore, the matrix of must be of the form
where at least one of