Show that multiplication by A maps every point in the plane onto the line.

Show that multiplication by $\displaystyle A = \begin{bmatrix}3 & 1\\ 6 & 2\end{bmatrix}$

maps every point in the plane onto the line $\displaystyle y = 2x$.

I tried to see what happens with the unit square and the points in the unit square is easy to see that they maps onto the line $\displaystyle y = 2x$. But does that equals that every point in the plane maps onto the line?

/regards

Re: Show that multiplication by A maps every point in the plane onto the line.

well, no, but: what can you say if A maps every basis element of a basis for $\displaystyle \mathbb{R}^2$ onto the line y = 2x?

conversely, given a point (t,2t) on the line, can you find some element (x,y) with A(x,y) = (t,2t)?