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Math Help - Independence, Dependence

  1. #1
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    Independence, Dependence

    HELP! I want to determine if these things are independent/dependent

    Don't know how to get the equations for these two:
    eg of what I mean by equation

    r(a, b) + r2(c, d) + r3(e, f) = 0
    ra + r2c + r3e = 0
    rb + r2d + r3f = 0


    1.
    {1, (sinx)^2, cos2x, (cosx)^2} in the F space, real fncts
    r1(1) + r2((sinx)^2) + r3(cos2x) + r4((cosx)^2) = 0

    from there I don't know how to group the terms so that I can set up a matrix and find out if trivial solutions exist.

    2.
    Similar problem as # 1 but it's with matrices in the M space of 2x2 matrices
    all of the following matrices are 2x2
    { matrix 1, matrix 2, matrix 3}

    [ 1 4 ] [ -1 5] [ -1 5]
    [ - 3 ] [ -1 5] [ -1 5]

    r1Matrix 1 + r2 Matrix 2 + r3 Matrix 3 = 0
    then elim to find triv.

    I just don't know how to set up the equations never seen examples.
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  2. #2
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    Quote Originally Posted by Noxide View Post
    HELP! I want to determine if these things are independent/dependent

    Don't know how to get the equations for these two:
    eg of what I mean by equation

    r(a, b) + r2(c, d) + r3(e, f) = 0
    ra + r2c + r3e = 0
    rb + r2d + r3f = 0


    1.
    {1, (sinx)^2, cos2x, (cosx)^2} in the F space, real fncts
    r1(1) + r2((sinx)^2) + r3(cos2x) + r4((cosx)^2) = 0


    \color{red}\mbox{Do you remember the Trigonometric Pythagoras Theorem }\cos^2x +\sin^2x=1?


    from there I don't know how to group the terms so that I can set up a matrix and find out if trivial solutions exist.

    2.
    Similar problem as # 1 but it's with matrices in the M space of 2x2 matrices
    all of the following matrices are 2x2
    { matrix 1, matrix 2, matrix 3}

    [ 1 4 ] [ -1 5] [ -1 5]
    [ - 3 ] [ -1 5] [ -1 5]


    \color{red}\mbox{The first one above is not a 2x2 matrix (most probably a typo), but never} \color{red}\mbox{ mind: can you see the 2nd and 3rd matrices are identical?}

    Tonio


    r1Matrix 1 + r2 Matrix 2 + r3 Matrix 3 = 0
    then elim to find triv.

    I just don't know how to set up the equations never seen examples.
    .
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  3. #3
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    Thanks Tonio

    Using trig identities I turned # 1 into:

    (cosx)^2 + (sinx)^2, (sinx)^2, (cosx)^2 -(sinx)^2, (cosx)^2

    Suppose I then do the following:
    r1[(cosx)^2 + (sinx)^2] + r2[(sinx)^2] + r3[(cosx)^2-(sinx)^2] + r4[(cosx)^2] = 0

    Is it then correct to do this:
    (cosx)^2 Terms: ( r1 + 0r2 + r3 + r4)
    (sinx)^2 Terms: ( r1 + r2 - r3 + 0r4)

    so that [( r1 + 0r2 + r3 + r4)(cosx)^2] + [( r1 + r2 - r3 + 0r4)(sinx)^2] = 0
    ie
    [( r1 + 0r2 + r3 + r4)(cosx)^2] =0
    [( r1 + r2 - r3 + 0r4)(sinx)^2] = 0
    and then create a linear system

    [1 0 1 1]
    [1 1 -1 0] r = 0

    which after row reduction becomes

    [1 0 1 1]
    [0 1 -2 -1]

    so we can then say that there are 2 pivot columns and 2 free columns therefore the presence of free columns suggests that the system has non-trivial solutions and therefore we can say that the vectors are indeed dependent


    With #2 I think I may need some help....

    I accidently wrote the incorrect matrices while in a rush to get to class. What I meant to write was:

    in the M space of 2x2 matrices are these 3 matrices independent or dependent?

    [ 1 4 ] [ -1 5] [ 1 13]
    [ -1 3 ] [ -1 5] [ 4 7]
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