Matrices 4: 2 × 2 Matrices and linear transformations

Overview:
  1. Introduction to matrices
  2. Adding and subtracting matrices
  3. Multiplying matrices
  4. 2 × 2 Matrices and linear transformations
  5. Determinants of 2 × 2 matrices
  6. Inverses of 2 × 2 matrices
  7. Invariant points and lines in 2 dimensions
  8. 3 × 3 Matrices and linear transformations
  9. Determinants of 3 × 3 matrices
  10. Inverses of 3 × 3 matrices
  11. Matrices and simultaneous equations

 Part 4: 2 × 2 Matrices and linear transformations


A 2 × 2 matrix can be used to apply a linear transformation to points on a Cartesian grid. A linear transformation in two dimensions has the following properties:

  • The origin (0,0) is mapped to the origin (it is invariant) under the transformation
  • Straight lines are mapped to straight lines under the transformation
  • Parallel lines remain parallel under the transformation
Questions
Which of the following are linear transformations?
  1. Translation by any non-zero vector
  2. Rotation about the origin by any angle
  3. Rotation about point P, by any angle greater than 0º but less than 360º, where P is not (0,0)
  4. Reflection in the y-axis
  5. Reflection in the line x = 0
  6. Reflection in the line y = mx where m is a constant
  7. Reflection in the line y = mx  + c where m and c are constants and c is non-zero
  8. Enlargement by any non-zero scale factor, centre of enlargement (0,0)
  9. Enlargement by any non-zero scale factor, centre of enlargement P, where P is not (0,0)
  10. Enlargement by scale factor 0, centre of enlargement (0,0)
Answers
  1. Translation by any non-zero vector is NOT a linear transformation because the origin is not mapped to itself.
  2. Rotation about the origin by any angle is a linear transformation.
  3. Rotation about point P, by any angle greater than 0º but less than 360º, where P is not (0,0)
  4. Reflection in the y-axis is a linear transformation.
  5. Reflection in the line x = 0 is a linear transformation.
  6. Reflection in the line y = mx where m is a constant is a linear transformation.
  7. Reflection in the line y = mx  + c where m and c are constants and c is non-zero is NOT a linear transformation because the origin is not mapped to itself.
  8. Enlargement by any non-zero scale factor, centre of enlargement (0,0) is a linear transformation.
  9. Enlargement by any non-zero scale factor, centre of enlargement P, where P is not (0,0) is NOT a linear transformation because the origin is not mapped to itself.
  10. Enlargement by scale factor 0, centre of enlargement (0,0) is NOT a linear transformation because straight lines aren’t mapped to straight lines; in fact every point on the grid is mapped to (0,0).

The effect of a 2 × 2 transformation matrix

To find where the matrix M \(\begin{pmatrix} a & b\\c & d\end{pmatrix}\) maps the point Q with coordinates \((x, y)\), we multiply the matrix M by the position vector representation of Q:

i.e. we do \(\begin{pmatrix} a & b\\c & d\end{pmatrix} \begin{pmatrix} x\\y\end{pmatrix} = \begin{pmatrix} x’\\y’\end{pmatrix}\), and Q is mapped to \((x’, y’)\).

For example, the matrix \(\begin{pmatrix} 2 & 1\\-1 & 3\end{pmatrix}\) maps \((1, 1)\) to \(\begin{pmatrix} 2 & 1\\-1 & 3\end{pmatrix} \begin{pmatrix} 1\\1\end{pmatrix} = \begin{pmatrix} 3\\2\end{pmatrix}\) or the point \((3, 2)\).

In the following applet, we will take a look at the effect of various transformations on the unit square OPQR:

Possible activities
  • Click on “Custom” towards the top of the applet in order to apply custom transformations to the unit square. Drag the blue slider fully to the right and tick the box to show the basis vectors. Then vary \(a\) and see the impact this has on the basis vectors. Then try varying \(b\), \(c\), and \(d\) (one at a time) to see the impact of varying these.
  • Demonstrate how the columns of the transformation matrix correspond to the transformations of two sides of the unit square given.
  • Drag the blue slider fully to the left.  Tick the boxes to show the basis vectors and to transform the gridlines too. Now drag the blue slider to the right. Note that on the transformed grid, the coordinates of the transformed shape are still at (0,0), (1,0), (1,1) and (0,1). The basis vectors in terms of the untransformed grid are however given by \(\begin{pmatrix} a\\c \end{pmatrix}\) and \(\begin{pmatrix} b\\d \end{pmatrix}\). This can be seen by unticking the “Transform gridlines too” box while the blue slider is fully dragged to the right.

Deducing transformation matrices for common transformations

In the applet above, the point P has position vector \(\begin{pmatrix} 1\\0\end{pmatrix}\) and the point R has position vector \(\begin{pmatrix} 0\\1\end{pmatrix}\). The transformation matrix \(\begin{pmatrix} a & b\\c & d\end{pmatrix}\) maps P to \(\begin{pmatrix} a\\c\end{pmatrix}\) and R to P to \(\begin{pmatrix} b\\d\end{pmatrix}\).

You can verify these by working out \(\begin{pmatrix} a & b\\c & d\end{pmatrix} \times \begin{pmatrix} 1\\0\end{pmatrix}\) and \(\begin{pmatrix} a & b\\c & d\end{pmatrix} \times \begin{pmatrix} 0\\1\end{pmatrix}\)respectively.

By visualising the unit square—in particular how a transformation affects the points P and Q—we can work backwards to quickly deduce the matrices representing many common transformations. For example, a rotation 90º anticlockwise about \((0,0)\) maps P to P’, with position vector \(\begin{pmatrix} 0\\1\end{pmatrix}\), and it maps R to R’ with position vector \(\begin{pmatrix} -1\\0\end{pmatrix}\). Therefore, the matrix representing this transformation is \(\begin{pmatrix} 0 & -1\\1 & 0\end{pmatrix}\).

Summary of transformation matrices that you should learn or be able to deduce quickly

Reflection in the \(x\)-axis: \(\begin{pmatrix} 1 & 0\\0 & -1\end{pmatrix}\)

Reflection in the \(y\)-axis: \(\begin{pmatrix} -1 & 0\\0 & 1\end{pmatrix}\)

Reflection in the \(y=x\): \(\begin{pmatrix} 0 & 1\\1 & 0\end{pmatrix}\)

Reflection in the \(y=-x\): \(\begin{pmatrix} 0 & -1\\-1 & 0\end{pmatrix}\)

Enlargement by scale factor \(k\), centre at \((0,0)\): \(\begin{pmatrix} k & 0\\0 & k\end{pmatrix}\)

Rotation 90º anticlockwise about \((0,0)\): \(\begin{pmatrix} 0 & -1\\1 & 0\end{pmatrix}\)

Rotation 180º \((0,0)\): \(\begin{pmatrix} -1 & 0\\0 & -1\end{pmatrix}\)

Rotation 270º anticlockwise about \((0,0)\): \(\begin{pmatrix} 0 & 1\\-1 & 0\end{pmatrix}\)

Rotation \(\theta\)º anticlockwise about \((0,0)\): \(\begin{pmatrix} \text{cos} \theta & -\text{sin} \theta\\ \text{sin} \theta & \text{cos} \theta \end{pmatrix}\)

Shear in the \(x\)-direction, shear factor \(k\): \(\begin{pmatrix} 1 & k\\0 & 1\end{pmatrix}\)

Shear in the \(y\)-direction, shear factor \(k\): \(\begin{pmatrix} 1 & 0\\k & 1\end{pmatrix}\)