How is [math]Mw[/math] is determined by the images of [math]e_1[/math] and [math]e_2[/math]?
We see that [math]Mw=M(ae_1+be_2)=M(ae_1)+M(be_2)=aM(e_1)+bM(e_2)[/math]. Just as [math]w[/math] is a linear combination of the standard basis vectors, [math]Mw[/math] is the [b]same[/b] linear combination of the [b]images[/b] of the standard basis vectors. [br][br]This is why the image of the integer lattice is a parallelogram lattice.