Overview:

Part 8: 3 × 3 matrices and linear transformations

Visualising transformations in 3D

3 × 3 matrices can be used to apply transformations in 3D, just as we used 2 × 2 matrices in 2D. To find where the matrix M \(\begin{pmatrix} a_{11} & a_{12} & a_{13}\\a_{21} & a_{22} & a_{23}\\a_{31} & a_{32} & a_{33}\end{pmatrix}\) maps the point Q with coordinates \((x, y, z)\), we multiply the matrix M by the position vector representation of Q:

i.e. we do \(\begin{pmatrix} a_{11} & a_{12} & a_{13}\\a_{21} & a_{22} & a_{23}\\a_{31} & a_{32} & a_{33}\end{pmatrix} \begin{pmatrix} x\\y\\z\end{pmatrix} = \begin{pmatrix} x’\\y’\\z’\end{pmatrix}\), and Q is mapped to \((x’, y’,z’)\).

For example, the matrix \(\begin{pmatrix} 2 & 1 & 0\\-1 & 3 & 0\\0 & 0 & 4\end{pmatrix}\) maps \((1, 1, 1)\) to \(\begin{pmatrix} 2 & 1 & 0\\-1 & 3 & 0\\0 & 0 & 4\end{pmatrix} \begin{pmatrix} 1\\1\\1\end{pmatrix} = \begin{pmatrix} 3\\2\\4\end{pmatrix}\) or the point \((3, 2, 4)\).

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

Deducing transformation matrices for common transformations

The transformation matrix \(\begin{pmatrix} a_{11} & a_{12} & a_{13}\\a_{21} & a_{22} & a_{23}\\a_{31} & a_{32} & a_{33}\end{pmatrix}\) maps \(\begin{pmatrix} 1\\0\\0\end{pmatrix}\) to \(\begin{pmatrix} a_{11}\\a_{21}\\a_{31}\end{pmatrix}\), \(\begin{pmatrix} 0\\1\\0\end{pmatrix}\) to \(\begin{pmatrix} a_{12}\\a_{22}\\a_{32}\end{pmatrix}\), and \(\begin{pmatrix} 0\\0\\1\end{pmatrix}\) to \(\begin{pmatrix} a_{13}\\a_{23}\\a_{33}\end{pmatrix}\).

You can verify these by working out \(\begin{pmatrix} a_{11} & a_{12} & a_{13}\\a_{21} & a_{22} & a_{23}\\a_{31} & a_{32} & a_{33}\end{pmatrix} \times \begin{pmatrix} 1\\0\\0\end{pmatrix}\), \(\begin{pmatrix} a_{11} & a_{12} & a_{13}\\a_{21} & a_{22} & a_{23}\\a_{31} & a_{32} & a_{33}\end{pmatrix} \times \begin{pmatrix} 0\\1\\0\end{pmatrix}\), \(\begin{pmatrix} a_{11} & a_{12} & a_{13}\\a_{21} & a_{22} & a_{23}\\a_{31} & a_{32} & a_{33}\end{pmatrix} \times \begin{pmatrix} 0\\0\\1\end{pmatrix}\) and respectively.

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

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

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

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

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

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

Rotation \(\theta\)º anticlockwise about the \((x\)-axis: \(\begin{pmatrix} 1 & 0 & 0\\ 0 & \text{cos} \theta & -\text{sin} \theta\\ 0 & \text{sin} \theta & \text{cos} \theta \end{pmatrix}\)

Rotation \(\theta\)º anticlockwise about the \((y\)-axis: \(\begin{pmatrix} \text{cos} \theta & 0& \text{sin} \theta\\ 0 & 1 & 0\\ -\text{sin} \theta &amp 0 & \text{cos} \theta \end{pmatrix}\)

Rotation \(\theta\)º anticlockwise about the \((z\)-axis: \(\begin{pmatrix} \text{cos} \theta & -\text{sin} \theta & 0\\ \text{sin} \theta & \text{cos} \theta & 0 \\ 0 & 0 & 1 \end{pmatrix}\)