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Math Help - Help with terminology and notation

  1. #1
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    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.
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  2. #2
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    Quote Originally Posted by JG89 View Post
    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.
    .
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  3. #3
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    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!
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