Number of linear transformations from a plane in $\Bbb R^3$ to a point in $\Bbb R^2$

Question

I'm really struggling with this question,

I have a plane in $\Bbb R^3$ with equation:

$x+y+z=1$

and a point \begin{bmatrix}1\\1\end{bmatrix} in $\Bbb R^2$.

How many linear transformations are there that send the plane into the point?

I would say infinite, the plane being an affine subspace is made up of 3 vectors, so i only need to send one of them to $[1,1]$ and the other 2 to $[0,0]$, is this correct?

What if the plane was $x+y+z=0$ ?

Answer 1

Hint: Try this simpler situation first: Let $P$ be the plane $\{z=1\}.$ How many linear transformations $T:\mathbb R^3 \to \mathbb R^2$ are equal to $(1,1)$ on $P?$ Show that if $T$ is such a linear transformation, then $T(e_1)= (0,0),$ $T(e_2)=(0,0),$ $T(e_3) = (1,1).$ So how many linear transformations do that?