# Representations of the real numbers

I am familiar with the matrix representation of the (lie) group of reals \begin{bmatrix}1&a\\0&1\end{bmatrix} with $a \in \mathbb{R}$. This is a really dumb thing I'm stuck on and I can't figure it out but what is the Lie algebra of this group? If I take the derivative and evaluate it at the identity I get \begin{bmatrix}0&1\\0&0\end{bmatrix} but when I exponentiate this to get back to the Lie group, it's clear that this is not the answer. I'm aware the the Lie Algebra representation is of the form \begin{bmatrix}0&a\\0&0\end{bmatrix}, but I can't for the life of me understand how you get that by differentiating near the identity? There's clearly something I'm missing.

Why is that not the answer? $$\exp \pmatrix{0 & a\cr 0 & 0\cr} = \pmatrix{1 & a\cr 0 & 1\cr}$$ as you should be able to check using whichever definition of the exponential you prefer.
• It is the case. The Lie algebra is spanned by the matrix you got by differentiation, $\pmatrix{0 & 1\cr 0 & 0\cr}$. – Robert Israel Aug 16 '16 at 0:20