## 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{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}

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.