If we interpret a two-dimensional vector v(x_{v};y_{v}) as a point V(x_{v};y_{v}) in the cartesian plane Oxy, the linear invertible operators represented by square 2x2 matrices act as bijective transformations of the point V(x_{v};y_{v}) into a point V'(x_{v'};y_{v'}) of Oxy. We shall call these transformations geometric linear transformations.
Such transformations show interesting geometric properties.
They transform the origin O into itself.
Given in Oxy the triangle τ with counterclockwise oriented vertices A(x_{A};y_{A}), B(x_{B};y_{B}), C(x_{C};y_{C}), the double area of τ is
A geometric linear transformation transforms the triangle τ into a triangle τ’ with double area
The determinant in the second side is the product of the determinants
therefore
A geometric linear transformation L transforms a triangle τ with area A into a triangle τ' with area A' given by the product between A and the determinant of L.
Let we consider two straight lines parallel each other but not parallel to any coordinate axis. Let the first intersects the axes in points U(p;0) and V(0;q), the second in points R(kp;0) and S(0;kq).
The linear transformation transforms these points into the following
The slopes of the transformed lines are equal.
Therefore a linear transformation transforms parallel straight lines into parallel straight lines.
A linear transformation transforms parallelograms into parallelograms.