# Thread: Help with terminology and notation

1. ## Help with terminology and notation

Two things:

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

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 $\mathbb{R}^{n^2}$ I am asked to prove that $df_{A_0} (X) = -X$ where $A_0$ is the identity matrix. My question is, $df_{A_0}$ would usually denote that derivative of f at the point $A_0$, so where does that (X) part come into play?

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.

2. Originally Posted by 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:

$(a_{11},a_{12},a_{21},a_{22})$ ...this is an element of $\mathbb{R}^4$ , 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 $\mathbb{R}^{n^2}$ I am asked to prove that $df_{A_0} (X) = -X$ where $A_0$ is the identity matrix. My question is, $df_{A_0}$ would usually denote that derivative of f at the point $A_0$, so where does that (X) part come into play?

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

3. TONIO!!!! That's it!! I've spent hours trying to figure out that freakin' notation and you're absolutely right. I totally forgot that my prof uses that notation for directional derivatives. Thanks so much!