A simple question about matrix groups. According to Geometrical Methods of Mathematical Physics (Schutz, p. 95),
The one-parameter subgroup generated by any matrix $A$ is the integral curve through [the identity] $e$ of the left-invariant vector field whose tangent at $e$ is $A$.
The author is referring to the general linear group in $n$ real dimensions, $GL(n,\mathrm{R})$.
So for $GL(n,\mathrm{R})$, the tangent vectors at $e$ are represented by matrices. But any vector $\overline{V}$ can be written in terms of a basis, say $$\overline{V}= V^1\frac{\partial}{\partial x^1}+V^2\frac{\partial}{\partial x^2} + ...$$
So how do we express a matrix, such as $\begin{bmatrix} a & b\\ c & d\\ \end{bmatrix}$, in terms of basis vectors $\frac{\partial}{\partial x^i}$? Wouldn't we need to express it with a rank-2 basis: $\frac{\partial}{\partial x^i}\otimes\frac{\partial}{\partial x^j}$?