Describe geometrically the set of all points in the plane with coordinates of the form m(0,1) + n(1,1), where m and n are integers
Well (0,1) and (1,1) form a basis for the entire plane, don't they?
You can describe any point in the two-dimensional plane using a linear combination of (0,1) and (1,1). eg. If you want to describe the point (16,7) using m(0,1) and n(1,1), you can let n = 16 and m = -9. Then adding the two you get -9*(0,1) + 16(1,1) = (0,-9)+(16,16) = (16,7).
You can form any (x,y) in the plane using appropriate m and n. So the set of points is the entire 2-dimensional plane... I hope that helps somewhat...
Well that does not answer the original question. Look at the question.
Now your reading may be the intended one. But it is not that stated one.
So the requested set is the set of lattice points in the plane that look like $\displaystyle \{(n,m+n):\{m,n\}\subset \mathbb{Z}\}$.