Matrix transformations

Examples of transformations

• Rotations
• Stretches
• Reflections
• Shears
• Translations - but these aren’t matrix transformations because they don’t keep the origin fixed.

Combining transformations

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

BA

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

• 90° clockwise rotation matrix:
  • \begin{pmatrix}0 & 1\\-1 & 0\end{pmatrix}
• Reflection in the line y=x matrix:
  • \begin{pmatrix}0 & 1\\1 & 0\end{pmatrix}
• Multiply them in reverse:
  • \begin{pmatrix}0 & 1\\1 & 0\end{pmatrix} \begin{pmatrix}0 & 1\\-1 & 0\end{pmatrix}
  • = \begin{pmatrix}0(0)+1(-1) & 0(1)+1(0)\\1(0)+0(-1) & 1(1)+0(0)\end{pmatrix}
  • = \begin{pmatrix}-1 & 0\\0 & 1\end{pmatrix}
• Answer: \begin{pmatrix}-1 & 0\\0 & 1\end{pmatrix}

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:

\begin{pmatrix}x\\y\end{pmatrix} \to \begin{pmatrix}...\\...\end{pmatrix}

or as a matrix multiplication:

\begin{pmatrix}a & b\\c & d\end{pmatrix} \begin{pmatrix}x\\y\end{pmatrix} = \begin{pmatrix}ax + by\\cx + dy\end{pmatrix}

Find a matrix to map \begin{pmatrix}x\\y\end{pmatrix} to \begin{pmatrix}2y+x\\3x\end{pmatrix}

• \begin{pmatrix}a & b\\c & d\end{pmatrix} \begin{pmatrix}x\\y\end{pmatrix} = \begin{pmatrix}2y+x\\3x\end{pmatrix}
• So we need to solve:
  • ax + by = 2y + x
  • cx + dy = 3x
• Solve ax + by = 2y + x:
  • a = 1
  • b = 2
• Solve cx + dy = 3x:
  • c = 3
  • d = 0
• Write this as a matrix:
  • \begin{pmatrix}1 & 2\\3 & 0\end{pmatrix}
• Answer: \begin{pmatrix}1 & 2\\3 & 0\end{pmatrix}

Find a matrix to map \begin{pmatrix}x\\y\end{pmatrix} to \begin{pmatrix}2y+x\\3x+y\end{pmatrix}

• \begin{pmatrix}a & b\\c & d\end{pmatrix} \begin{pmatrix}x\\y\end{pmatrix} = \begin{pmatrix}2y+x\\3x+y\end{pmatrix}
• Equations we can form:
  • ax+by=2y+x
  • cx+dy=3x+y
• Solve ax + by = 2y + x:
  • ax=1x \Rightarrow a=1
  • by=2y \Rightarrow b=2
• Solve cx + dy = 3x + y:
  • cx=3x \Rightarrow c=3
  • dy=y \Rightarrow d=1
• Write this as a matrix:
  • \begin{pmatrix}1 & 2\\3 & 1\end{pmatrix}
• Answer: \begin{pmatrix}1 & 2\\3 & 1\end{pmatrix}

Find the coordinates of the points (1,1), (3,1), (3,3) and (1,3) after a transformation by matrix \begin{pmatrix}-1 & 2 \\ 2 & 1\end{pmatrix}

• Find a column matrix for all the points combined:
  • Write the corresponding coordinates in each column
  • \begin{pmatrix}1 & 3 & 3 & 1\\1 & 1 & 3 & 3\end{pmatrix}
• Multiply the transformation matrix by the column matrix (make sure you do it in the right order, transformation matrix first):
  • \begin{pmatrix}-1 & 2 \\ 2 & 1\end{pmatrix} \begin{pmatrix}1 & 3 & 3 & 1\\1 & 1 & 3 & 3\end{pmatrix}
  • = \begin{pmatrix}-1(1)+2(1) & -1(3)+2(1) & -1(3)+2(3) & -1(1)+2(3)\\2(1)+1(1) & 2(3)+1(1) & 2(3)+1(3) & 2(1)+1(3)\end{pmatrix}
  • = \begin{pmatrix}1 & -1 & 3 & 5\\3 & 7 & 9 & 5\end{pmatrix}
• Answer(s):
  • (1,3)
  • (-1,7)
  • (3,9)
  • (5,5)

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 1 or -1

Transformations which do not change the size of shapes (e.g. rotations and reflections) have a determinant of either 1 or -1.

• A determinant of 1 means the shape keeps its orientation (e.g. a rotation) and its size the same.
• A determinant of -1 means the shape changes its orientation (e.g. a reflection) but keeps its size the same.

Singular matrix

A singular matrix has a determinant of 0, which means that it will create an image which has no area. For example:

• A line
• A single point (for a transformation of \begin{pmatrix}0 & 0\\0 & 0\end{pmatrix})

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.

(X)(X)^{-1} = I

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