Linear transformations - reflections

Linear transformations - reflections

Linear Transformations: Reflections

Basics of Reflection

  • A reflection is a type of linear transformation where a figure is flipped across a line, creating a mirror image.
  • Linear transformations, including reflections, can be represented by matrices, where multiplication of a matrix by a vector results in a transformed vector.
  • The line of reflection can be across a coordinate axis, such as the x or y axis, or any other line in the plane.

Reflection in the X-axis

  • A reflection in the x-axis keeps the x-coordinate of each point the same and multiplies the y-coordinate by -1.
  • This can be represented by the 2x2 matrix [[1,0],[0,-1]].

Reflection in the Y-axis

  • A reflection in the y-axis multiplies the x-coordinate by -1, while keeping the y-coordinate the same.
  • This can be represented by the 2x2 matrix [[-1,0],[0,1]].

Reflection in the Line y=x

  • A reflection in the line y=x interchanges the x and y coordinates of each point.
  • This can be represented by the 2x2 matrix [[0,1],[1,0]].

Reflection in the Line y=-x

  • A reflection in the line y=-x interchanges the x and y coordinates of each point and changes their signs.
  • This can be represented by the 2x2 matrix [[0,-1],[-1,0]].

Properties of Reflections

  • Reflection is a reversible process. Reflecting a figure twice across the same line will return the figure to its original position.
  • The reflection of a figure across line A and then line B is the same as a single reflection across line C, where line C is the perpendicular bisector of the angle between lines A and B.

This is a brief overview of the fundamentals of reflections in linear transformations. Applying these transformations to different types of shapes and analysing the results will enable you to better understand their properties and effects.