Projective coordinates vs vectors

[Stephen Tashi]

There is a technical distinction between a vector and the coordinates of a vector. Are projective (also called "affine") coordinates the coordinates of vectors?

I'm thinking of how translation is accomplished by matrix multiplication. For example the point $(x,y)$ in 2-D is given coordinates $(x,y,1)$ and translation by $(h,v)$ is represented as:
$$\begin{pmatrix} 1&0&h \\ 0&1&v \\ 0&0&1 \end{pmatrix} \begin{pmatrix} x \\ y\\ 1 \end{pmatrix} = \begin{pmatrix} x+h \\ y+v \\ 1 \end{pmatrix}$$.

Students are told that matrix multiplication performs a linear transformation on a vector space and also disturbed by the exercise showing that translation by a (non-zero) vector is not a linear transformation . What are the saving legalisms here?

 

[Office_Shredder]

Projective and affine coordinates are definitely not the same thing. Projective space (in 2 dimensions let's say) is the set of all points [x:y:z] where [a:b:c] and [x:y:z] are considered equivalent if there is some number k such that ak=x, bk=y, ck=z.

It is the case that if you restrict to the set of [x:y:1] that there is an obvious bijection between these points and points of the form (x,y) with no equivalence relation - these (x,y) coordinates are what is called affine space.

At first glance that matrix multiplication appears to make translation a linear operation, but the set of points of the form (x,y,1) is not a vector space in any way that will make that matrix multiplication a linear transformation. For example it is tempting to just declare (x,y,1)+(a,b,1) = (a+x,y+b,1) but then if the matrix is T, we no longer get that T(x,y,1) + T(a,b,1) = T(a+x,y+b,1) since the left hand side will be shifted by twice what the right hand side is shifted by.

 

[Bacle2]

When I taught an intro class in Linear Algebra, I told the students that , while innacurate, any expression of x of the type ax+b is linear, and that the linearity has to see with the fact that x is raised to the first degree. When someone pinned me down and wanted more of an explanation , I said that we were actually dealing with affine maps, which are the composition of a translation and a linear map. But I agree with you, it is confusing.