2.1.2 Matrix Dimensions

The size or dimension of a matrix is defined by the number of rows and columns it contains. If a matrix has $m$ rows and $n$ columns, it is referred to as an $m × n$ matrix.

Matrices in 3-D

The 3D identity matrix is:

$$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$

Any 3D linear transformation can be represented by a matrix in which the three columns are the images of the points $\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$, and $\begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}$, respectively.

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3D Reflection Matrices

When reflecting in 3D, we reflect in a plane rather than a line:

• $x = 0$ is also called the $y$-$z$ plane
• $y = 0$ is also called the $x$-$z$ plane
• $z = 0$ is also called the $x$-$y$ plane A diagram illustrating the planes is shown below:

• Red: $x = 0$ or $y$-$z$ plane
• Yellow: $y = 0$ or $x$-$z$ plane
• Blue: $z = 0$ or $x$-$y$ plane

Reflection Matrices

Reflection matrices transform vectors by flipping their coordinates relative to a given plane. The unaffected coordinates remain the same, while the reflected coordinate changes sign.

Reflection in the $z = 0$ plane

Here, only the $z-coordinate$s change, while the $x$ and $y$ coordinates remain unchanged.

Reflection Matrices for Other Planes

Reflection in the $x = 0$ plane:

Here, only the $x-coordinate$ changes sign.

$$\text{Ref}_x = \begin{pmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$

Reflection in the $y = 0$ plane:

Only the $y-coordinate$ changes sign.

$$\text{Ref}_y = \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$

Rotation Matrices

Rotation matrices describe transformations where a vector is rotated about a specific axis by an angle $\theta$.

Rotation About the $x-axis$

$$R_x = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos\theta & -\sin\theta \\ 0 & \sin\theta & \cos\theta \end{pmatrix}$$

Rotation About the $y-axis$

$$R_y = \begin{pmatrix} \cos\theta & 0 & \sin\theta \\ 0 & 1 & 0 \\ -\sin\theta & 0 & \cos\theta \end{pmatrix}$$

Rotation About the $z-axis$

$$R_z = \begin{pmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{pmatrix}$$