Lie groups and vector fields

Question

Suppose we have the group of affine transformations of the real line.

Our group consists of all $2\times2$ matrices of the form: $\begin{bmatrix}a&b\\0&1\end{bmatrix}$ where $a$ and $b$ are real numbers and $a>0$.

What is the way to find the basis for left-invariant vector fields?

Thanks.

Answer 1

Let $g=\begin{bmatrix} a&b\\0&1\end{bmatrix}$. The left-invariant $1$-forms are given by the entries of the matrix $g^{-1}dg = \begin{bmatrix} da/a & db/a \\ 0&0\end{bmatrix}$. What are the dual vector fields?

Alternatively, find the Lie algebra $\mathfrak g$ (i.e., $T_IG$), take an obvious basis, and left-translate it.