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Matrix transformations

Examples of transformations

Section titled “Examples of transformations”

Combining transformations

Section titled “Combining transformations”

When we combine transformations (e.g. do a transformation by matrix followed by a transformation by matrix ), we can represent the combined transformation by multiplying the matrices in reverse:

This is exactly the same as composite functions.

Find the single matrix which represents a 90° clockwise rotation followed by a reflection in the line y=x

Section titled “Find the single matrix which represents a 90° clockwise rotation followed by a reflection in the line y=x”
  • 90° clockwise rotation matrix:
  • Reflection in the line matrix:
  • Multiply them in reverse:
  • Answer:

Linear transformations

Section titled “Linear transformations”
  • A linear transformation is essentially a 2D transformation.
  • If a shape has straight edges, a linear transformation will keep them straight.

We can write a linear transformation either as:

or as a matrix multiplication:

Find a matrix to map to

Section titled “Find a matrix to map to ”
  • So we need to solve:
  • Solve :
  • Solve :
  • Write this as a matrix:
  • Answer:

Find a matrix to map to

Section titled “Find a matrix to map to ”
  • Equations we can form:
  • Solve :
  • Solve :
  • Write this as a matrix:
  • Answer:

Find the coordinates of the points , , and after a transformation by matrix

Section titled “Find the coordinates of the points , , and after a transformation by matrix ”
  • Find a column matrix for all the points combined:
    • Write the corresponding coordinates in each column
  • Multiply the transformation matrix by the column matrix (make sure you do it in the right order, transformation matrix first):
  • Answer(s):

Determinant and area scale factor

Section titled “Determinant and area scale factor”

The determinant of the transformation matrix gives the area scale factor of the transformation.

  • For example, If the determinant is 2, the area of any shape will double after the transformation.

Determinant of or

Section titled “Determinant of or ”

Transformations which do not change the size of shapes (e.g. rotations and reflections) have a determinant of either or .

  • A determinant of means the shape keeps its orientation (e.g. a rotation) and its size the same.
  • A determinant of means the shape changes its orientation (e.g. a reflection) but keeps its size the same.

Singular matrix

Section titled “Singular matrix”

A singular matrix has a determinant of , which means that it will create an image which has no area. For example:

  • A line
  • A single point (for a transformation of )

Self-inverse matrix transformations

Section titled “Self-inverse matrix transformations”

With transformations, we can pre-multiply by the inverse of a transformation matrix to reverse the transformation.

You need to do the inverse operations in the opposite order to how you did the transformations originally.