Problem
Problem 4.39 Using the coordinate vector fields $\partial / \partial x_{j}^{i}, 1 \leqslant i, j \leqslant n$, on GL( $\left.n, \mathbb{R}\right)$, prove that the vector field $Y$ on $\operatorname{GL}(n, \mathbb{R})$ whose matrix of components at the identity is $A=\left(a_{j}^{i}\right)$ and whose matrix of components is equal to $B A$ at the element $B=\left(b_{j}^{i}\right)$ of $\operatorname{GL}(n, \mathbb{R})$ is a left-invariant vector field.