\[x=(x_1 , x_2 ,x_3), \: y=(y_1,y_2,y_3), \: z=(z_1,z_2,z_3) \in \mathbb(R)^3\]
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.