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Math Help - Find the Basis

  1. #1
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    Find the Basis

    for the each of the following vector spaces V, find the Basis and give the dimension:
    a) V = [ 3x3 real matrices]
    b) V= smooth real-valued functions on [0,1] satisfying y'''-y' = 0]
    c) V= [real valued functions on a set X containing n points]
    d) V= [complex 2x2 matrices A for which Atr =A (where A is the matrix of complex conjugates) ]

    Attempt:

    a) standard basis with dim 9
    b)I actually don't know where to start on this one!
    c) a wild guess that dim v = n but again i'm struggling to think of the entries for the basis
    d) unsure but I have had a guess and I think it is dim 3 with these entries
    a+ib c+id a-ib c-id a+ib c-id
    c+id a+ib c-id a-ib c-id a+ib


    I appreciate any help I recieve. Thank you.
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  2. #2
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    Not sure, but...

    You can make bases out of the following:

    1. 9 matrices, each 3x3, each with a 1 in one of the matrix element slots and zeroes otherwise. I assume this is what you mean by the "standard" set.

    2. Either y'=0, or y''=y with y' not equal to 0. If y'=0, a basis is the function 1 spanning the set of y= constant functions. If y''=y, you have linear combinations of exp(x) and exp(-x), so a basis could be these 2 functions.

    3. Define a function to be a set of N points, with the m'th point being the value of the function on the m'th point in the set. Then a basis would be the standard N-component unit vectors (1 in one position, 0 otherwise).

    4. Are you working over C or over R? The 2x2 complex matrices have 8 degrees of freedom if over R, and 4 if over C - without your additional condition. The condition A transpose = A complex conjugate imposes the condition that the diagonal elements are real, and the off diagonal elements are related - just write out both of the matrices like you did in the problem and see what the condition implies in terms of the matrix elements.
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  3. #3
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    Quote Originally Posted by qmech View Post
    You can make bases out of the following:

    1. 9 matrices, each 3x3, each with a 1 in one of the matrix element slots and zeroes otherwise. I assume this is what you mean by the "standard" set.

    2. Either y'=0, or y''=y with y' not equal to 0. If y'=0, a basis is the function 1 spanning the set of y= constant functions. If y''=y, you have linear combinations of exp(x) and exp(-x), so a basis could be these 2 functions.
    So for the entire problem, a basis consists of \{1, e^{x}, e^{-x}\}.

    3. Define a function to be a set of N points, with the m'th point being the value of the function on the m'th point in the set. Then a basis would be the standard N-component unit vectors (1 in one position, 0 otherwise).

    4. Are you working over C or over R? The 2x2 complex matrices have 8 degrees of freedom if over R, and 4 if over C - without your additional condition. The condition A transpose = A complex conjugate imposes the condition that the diagonal elements are real, and the off diagonal elements are related - just write out both of the matrices like you did in the problem and see what the condition implies in terms of the matrix elements.
    Last edited by HallsofIvy; January 13th 2010 at 03:38 AM.
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  4. #4
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    Quote:
    3. Define a function to be a set of N points, with the m'th point being the value of the function on the m'th point in the set. Then a basis would be the standard N-component unit vectors (1 in one position, 0 otherwise).

    I am unsure with regards to 3.
    does this mean dim V = n?


    Thank you for you help above, I've got 1 and 2 now fully understood, working on 4 now but I think I understand it from above.
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