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...