# Transformations of the Unit Square in the x - y Plane

## Transformations of the Unit Square in the x - y Plane

## Identifying Transformations

- Understand that transformations can be recognised by the image of the unit square under the transformation.
**Scaling**is marked by stretching or shrinking of the unit square and is caused by a scalar factor in the matrix.**Rotation**is recognised when the unit square is reoriented but maintains the same size and shape.- A
**reflexion**is notable if the image is a mirror version of the unit square. - A
**shear**is typified by the square appearing to be pushed to one side.

## Properties of the Unit Square

- Note that the unit square in the x-y plane motivates the concept of any square with sides parallel to the axes.
- Be aware that the unit square refers to a square of side length 1, whose bottom-left corner is at the origin.
- Remember that
**transformations**can be applied to the unit square to help visualise matrix transformations.

## Working with Transformations

- Understand that transformation of the unit square can be represented using
**matrices**to change the position of the points. - Learn to multiply the coordinates of the square by the matrix to find the transformed coordinates. This applies to transformations including enlargement, reflections, rotations and shearing.
- Be confident manipulating the unit vectors
**i**(pointing in the direction of one unit on the x-axis) and**j**(pointing in the direction of one unit on the y-axis).

## Analysing the Impact of Transformations

- Understand that an unchanged unit square signifies the
**identity matrix**. - A negative determinant of the transformation matrix indicates a reflexion has occurred.
- Keep in mind that the
**determinant**of a transformation matrix gives the area scale factor. - Remember, if the determinant is 1 or -1, the transformation is area-preserving (i.e., the area of the unit square remains the same).