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: $$\begin{pmatrix} \text{cos} \theta & -\text{sin} \theta & 0\\ \text{sin} \theta & \text{cos} \theta & 0 \\ 0 & 0 & 1 \end{pmatrix}$$