3x3 Invertible transformations

[Aleoa]

Homework Statement

\mathbb{P}^{2} is an affine plane of 2 dimensions

The Attempt at a Solution

Take for example the affine plane with z=1. Then I take a general vector v= [x,y,1] and i apply the transformation B and then the transformation A.
So i get Bv=f(v) and Av=cf(v).

To me this doesn't seems the same transformation in the affine plane.
Where am i wrong ?

 

[fresh_42]

So i get Bv=f(v) and Av=cf(v).

To me this doesn't seems the same transformation in the affine plane.
Where am i wrong ?

You're not, but we have the projective plane here, not the affine.
Let $f(v)=(f_1:f_2:f_3)$, then projective means $(f_1:f_2:f_3) = (c \cdot f_1:c \cdot f_2:c \cdot f_3)$. This is exactly what projective means: the relations between the coordinates are equal, not the coordinates themselves.

 

[Ray Vickson]

Homework Statement

View attachment 224549

\mathbb{P}^{2} is an affine plane of 2 dimensions

The Attempt at a Solution

Take for example the affine plane with z=1. Then I take a general vector v= [x,y,1] and i apply the transformation B and then the transformation A.
So i get Bv=f(v) and Av=cf(v).

To me this doesn't seems the same transformation in the affine plane.
Where am i wrong ?

Please take the trouble to actually type out the problem statement; your attached image is unreadable on my devices. Read the post "Guidelines for students and helpers" for more about this issue!