The representation space of the fundamental representation of $SE(3)$

The fundamental rep $\rho$ of an element $g\in SE(3)$ is

$$\rho (g)=\left(\begin{array}{cc}1& 0^T\\ \mathbf{t}& R\end{array}\right)$$

Curiously, the fundamental rep is a 4x4 matrix, despite the fact that it acts on a 3-dimensional space ${\mathbb{E}}^3$. This is due to the fact that $SE(3)$ includes translations, and translations can only be represented as linear transformations in what’s called ‘homogeneous coordinates’ (recall that the definition of a representation are a set of linear transformations that have the same group structure as the original group). That is, given $\mathbf{r}\in {\mathbb{E}}^3$

$$g\cdot \mathbf{r}=\rho (g)\left(\begin{array}{c}1\\ \mathbf{r}\end{array}\right)=\left(\begin{array}{c}1\\ R\mathbf{r}+\mathbf{t}\end{array}\right)$$

This is where the '$1$' comes from.

We can similarly compute the action of $g$ on a tangent vector $\mathbf{d}\in T{\mathbb{E}}^3$, which we will denote as $\hat{\cdot }$, by replacing the $1$ with a $0$.

$$g\hat{\cdot }\mathbf{d}=\rho (g)\left(\begin{array}{c}0\\ \mathbf{d}\end{array}\right)=\left(\begin{array}{c}0\\ R\mathbf{d}\end{array}\right)$$

We can simulatenously compute the action of $g$ on many elements of ${\mathbb{E}}^3$ and $T{\mathbb{E}}^3$ like so

$$\begin{array}{rl}\rho (g)\left(\begin{array}{ccccc}1& 1& 0& 1& 0\\ {\mathbf{r}}_1& {\mathbf{r}}_2& {\mathbf{d}}_1& {\mathbf{r}}_3& {\mathbf{d}}_2\end{array}\right)& \\ =\left(\begin{array}{ccccc}1& 1& 0& 1& 0\\ R{\mathbf{r}}_1+\mathbf{t}& R{\mathbf{r}}_2+\mathbf{t}& R{\mathbf{d}}_1& R{\mathbf{r}}_3+\mathbf{t}& R{\mathbf{d}}_2\end{array}\right)& \end{array}$$