I'm reading this text on computer graphics, and came across the following section.
The 4x4 homogenous matrix I'm accustomed to looks like this
$\begin{bmatrix} r_{11} & r_{12} & r_{13} & t_{14} \\ r_{21} & r_{22} & r_{23} & t_{24} \\ r_{31} & r_{32} & r_{33} & t_{34} \\ 0 & 0 & 0 & 1 \end{bmatrix}$
Where the $r_{11}$ through $r_{33}$ define a 3x3 rotation matrix, and $t_{14}$, $t_{24}$, and $t_{34}$ define a 3x1 translation vector.
I'm also familiar with the "change of coordinates" matrix. Suppose we have vector spaces A and B in $\mathbb{R}^3$. I can use the following matrix to take a vector in B, to a vector in A.
$P_{A \leftarrow B} = \begin{bmatrix} [b_1]_A & [b_2]_A & [b_3]_A\end{bmatrix}$
where $[b_i]_A$ is the i-th basis vector of B in A-coordinates.
Given this information, I'm trying to interpret how the rows of a 4x4 matrix represent the x-axis, y-axis, z-axis, and translation.