## 6. Some interesting linear transformations.

• A geometric linear transformation transforms the point P(x;y) into the point P'(-x;y): Sy is a symmetry with respect to the ordinate axis.
• A geometric linear transformation transforms the point P(x;y) into the point P'(x;-y): Sx is a symmetry with respect to the abscissa axis.
• A geometric linear transformation transforms the point P(x;y) into the point P'(kx;ky): S is a homothety with center at O.

• H multiplies by k the lengths of the segments: k is the homothety ratio.
• In particular, if k=1 H is the identity, if k = -1 H is a symmetry with respect to O.
• Homotheties do not change the slopes of the transformed straight lines, therefore the homothetic polygons have angles ordinately congruent.
• A geometric linear transformation rotates the point P(x;y) by an angle α: R is a rotation with center at O.

Rotations transform the segments into segments with equal length: therefore they are isometries.

• The composition of a rotation with a homothety is a similarity with center at O: