TITLE: In what sense can the Lie algebra of $GL_n(\mathbb{C})$ be "identified" with $M_n(\mathbb C)$?
QUESTION [6 upvotes]: I would like to do Tao's exercise 6 (i) but before I can even attempt it I need to be clear about his terminology.
Exercise 6
Show that the Lie algebra $gl_n(\mathbb{C})$ of the general linear group $GL_n(\mathbb{C})$ can be identified with the space $M_n(\mathbb{C})$ of $n \times n$ complex matrices, with the Lie bracket $[A,B] := AB - BA$.
When he writes "identified", does he just mean there is an isomorphism? So the exercise would be to find an explicit isomorphism from $M_n(\mathbb{C})$ to $gl_n(\mathbb{C})$? Is that correct?
Many thanks for your help.

REPLY [2 votes]: "Identify" usually means "exhibit a nice isomorphism."  Making that rigorous in general is a bit tricky, but you know it when you see it. 
In this case, you could make it rigorous in the following way.  Note that both $\mathfrak{gl}_n(\mathbb{C})$ and $M_n(\mathbb{C})$ have a natural action of $\mathrm{GL}_n(\mathbb{C})$ (by the adjoint action and conjugation respectively), so a "nice" (or "natural" or "canonical") isomorphism should respect this action.
But Tao is almost certainly using "identify" informally.  He just means find an isomorphism, with the subtext that you shouldn't need to make any "arbitrary choices" along the way.