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The Projective Linear Transformation

Suppose we have points  
\[x=(x_1 , x_2 ,x_3), \: y=(y_1,y_2,y_3), \: z=(z_1,z_2,z_3) \in \mathbb(R)^3\]
  and the operation  
\[T(x,y,z) \rightarrow (x,y)\]

Then  
\[T\]
  is a linear transformation is called a projective transformation.
It is linear because
\[\begin{equation} \begin{aligned} T(\alpha x + \beta x' , \alpha y + \beta y' , \alpha z + \beta z') &= (\alpha x + \beta x' , \alpha y + \beta y') \\ &= (\alpha x, \alpha y)+( \beta x' + \beta y') \\ &=\alpha (x,y) + \beta (x',y') \\ &=\alpha T(x,y,z) +\beta (x',y'z') \end{aligned} \end{equation}\]

This particular example projects the  
\[\{ (x,y,z) \in \mathbb{R}^3 \}\]
  space onto the subspace  
\[\{(x,y) \in \mathbb{R}^2 \}\]
.
Because  
\[T(x,y,z)=(x,y,0)\]
  the image of the transformation is the  
\[xy\]
  plane and  
\[T(x,y,z)=(0,0,0)\]
 whatever the value of  
\[z\]
  so the kernel of the transformation is the line along the  
\[z\]
  axis.

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