Linear maps

[TsAmE]

Let S: \(\displaystyle \mathbb{R}^2 \to \mathbb{R}^2\) be the function defined by S(x, y) = (x - y, y) for all (x,y)\(\displaystyle \varepsilon \mathbb{R}^2\)

What is the image under S of the vertical line x = b?

Attempt:

Since S(b, y) = (b - y, y)

therefore the x will be a constant and the x coordinate will be shifted downwards.

However the correct answer was that the y-axis (the line x = 0) is taken to the line y = -x. All other vertical lines are
taken to lines that are parallel to this line.

I dont understand this

 

[prex]

I think you'd better use new coordinates system like (x1,y1).
x1=x-y, y1=y
So, if your initial region is x=b, the region after transformation will be like that: x1=b-y, y1=y.
If you choose b=0, you obtain x1=-y1