What is the purpose of a transformation matrix in geometry?

The purpose of a transformation matrix in geometry is fundamentally about manipulating coordinates and altering figures. A transformation matrix allows you to perform operations such as translation, rotation, scaling, and shearing on geometric shapes. By applying a transformation matrix to a set of coordinates in space, you can change their position, size, or orientation without changing the intrinsic properties of the shapes, offering a powerful tool in computational geometry and graphics.

For instance, when scaling a figure, a transformation matrix will adjust the coordinates according to a specified scale factor, effectively enlarging or reducing the size of the geometric figure. Similarly, rotation matrices reposition a figure around a specified point by adjusting the angles of the coordinates based on trigonometric principles. Therefore, the ability to change coordinates or scale figures succinctly captures the effectiveness and purpose of transformation matrices in geometry.