Linear transformation understanding

Question

Let $S: R^2 → R^2$ be the function defined by $S(x, y) = (x − y, y)$ for all $(x, y)$ I've found the matrix for this which was question 1.

These are the two questions I am struggling to understand.

(1) What does S do to horizontal lines of the form $y = a$ ?

(2) What does S do to vertical lines of the form $x = b$ ?

For (1) I can see that the line $y=a$ is projected to itself but there was something else in the memo which said that points on the x-axis are left unchanged. This is what I don't understand because for the transformation for the x co-ordinate it would be $x=x-a$. This would clearly no longer be on the x-axis.

I think what I really need is to understand how these linear maps affect the mapping of these line, and how to interpret the mapping with variables.

Answer 1

1. Take a point $(x,a)$ on the line $y=a$. It becomes $(x-a,a)$. That is, the line shifts left by $a$ units (still the same line).
2. Take a point $(b,y)$ on the line $x=b$. It becomes $(b-y,y)=(b,0)+y(-1,1)$. The line rotates $45$ degrees positive around $(b,0)$ (draw the picture to see it).