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**JG89** Two things:

1) If we say that the space of all 2 by 2 matrices is identified with R^4, what does that mean?

Write both rows of a matrix one after the other with commas between the entries and enclosed in parentheses:

$\displaystyle (a_{11},a_{12},a_{21},a_{22})$ ...this is an element of $\displaystyle \mathbb{R}^4$ , and this is what is meant, since the other direction if also clearly true.

2) Suppose f is a function from GL(n, R) to GL(n, R) (the space of all real n by n invertible matrices) identified with $\displaystyle \mathbb{R}^{n^2} $ I am asked to prove that $\displaystyle df_{A_0} (X) = -X $ where $\displaystyle A_0 $ is the identity matrix. My question is, $\displaystyle df_{A_0} $ would usually denote that derivative of f at the point $\displaystyle A_0 $, so where does that (X) part come into play?

I don't know this one, but I think the notation $\displaystyle df_{A_0}(X)$ may denote directional derivative of $\displaystyle F(X)$ in the direction of $\displaystyle A_0$ ...I can't say

Tonio

I know that I should be asking my prof this, but I wanna do these homework questions before my next class (Wednesday), so it would be great if you guys could help me out.