Homogeneous Coordinate in Linear Algebra Class

Question

The line y=2x +1 can be expressed in vector form as ??[1 2]+[0 1] (let x=t, then y=2t+1).

 

A) (using homogeneous coordinates) what is the matrix of a translation of 2 in the x direction and 3 in the y direction?

 

B) Show that the image of the line y=2x+1 under this translation is the line y=2x. {After matrix multiplication, the image will be in vector form. You have to show that “the y part = 2 times the x part”}

Answer 1

I agree with Mr. G's concept however one can use matrix multiplication in Homogenous Coordinates. The points are

P1=[0 1 1] and P2=[1 3 1] transposed. The 3x3 translation matrix is say A =  [1 0 2; 0 1 3; 0 0 1]. Then A*P1 =

[2, 4, 1] transpose and A*P2=[3 6 1] transpose. In the cartesian frame (2,4), and (3,6). Then y=2*x in both cases or the y part = 2 times the x part. The matrix is given in the literature. 

Answer 2

I may be missing something here, but in part B), the line y = 2x + 1 contains the points (0,1) and (1,3). Under the given translation

 

(x,y) → (x+2,y+3),

 

these points move to (0,3) and (3,6), which both lie on the line y = 2x. 

 

If you want to do this with matrix multiplication, you need a matrix for the translation. Since the translation is NOT a linear transformation (it is called affine), I am not sure how you can find a matrix representing it.

 

Could you possibly reread the problem and verify that you are using the correct terms in part A)? Specifically, does the problem state "translation" as opposed to something else, like "shearing?" Shears are linear transformations, while translations aren't.

 

Please help me with this. There is some confusing information here.