Understanding Matrices and GL(n,R) Functions

[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?

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 want to do these homework questions before my next class (Wednesday), so it would be great if you guys could help me out.

 

[mathman]

For the first question: a (real) 2 by 2 marix is specified by four numbers, which defines a point in R^4.

Second question: I am not familiar with the notation.

 

[Office_Shredder]

For the second question: The derivative of a function is a linear function. In this case the question is asking you to prove that the linear function that the derivative is is the function df(X)=-X

 

[JG89]

OfficerShredder, I was thinking that. But usually my prof would use the notation $$df_{A_0}$$ to denote the derivative of f at $$A_0$$. Why add in the extra $$(X)$$ ?

And mathman, do I read off the entries of the matrix row by row or column by column

 

[JG89]

In case it helps, OfficerShredder, f is defined by $$f(A) = A^{-1}$$ if $$A \in GL(n,r)$$

 

[Landau]

And mathman, do I read off the entries of the matrix row by row or column by column

That's up to you. You can identify the space of nxm-matrices with R^{mn} in a lot (namely (nm)!) of ways, there's not really a preferred way.

 

[HallsofIvy]

OfficerShredder, I was thinking that. But usually my prof would use the notation $$df_{A_0}$$ to denote the derivative of f at $$A_0$$. Why add in the extra $$(X)$$ ?

And mathman, do I read off the entries of the matrix row by row or column by column

For the same reason that to talk about the "squaring function" we say $f(x)= x^2$ rather than just "$f= ( )^2$". A function is defined by what it does to values of x.

 

[JG89]

I'm still not getting it. In my prof's usual notation $$df_{A_0}$$ would mean the derivative of f at $$A_0$$. If you write it using the prime notation, $$df_{A_0} = f'(A_0)$$. I still don't see why you would need the matrix $$X$$ when we're evaluating the derivative function at the point $$A_0$$

 

[JG89]

lol nevermind guys. I totally forgot that my proof uses that notation to mean the directional derivative of f at the point X with respect to A_0