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Math Help - Linearly independent sets

  1. #1
    Senior Member chella182's Avatar
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    Linearly independent sets

    Are the following sets of functions linearly independent or not? Explain your answer.

    a) f(X):= \sin{(2X)}+e^X,f'(X),f''(X),f'''(X).
    b) f(X):=X^3+1,f(X)^2,f(X)^3
    c) \cos{(2x)},\sin{(X)}^2,\cos{(X)}^2

    Literally no clue here, our teacher for this module is shockingly bad. I would ask you not to be vague in your answers, but I'm asking for help it would be a little rude, but yeah... assume I know little to nothing about this topic
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  2. #2
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    Quote Originally Posted by chella182 View Post
    Are the following sets of functions linearly independent or not? Explain your answer.

    a) f(X):= \sin{(2X)}+e^X,f'(X),f''(X),f'''(X).
    b) f(X):=X^3+1,f(X)^2,f(X)^3
    c) \cos{(2x)},\sin{(X)}^2,\cos{(X)}^2

    Literally no clue here, our teacher for this module is shockingly bad. I would ask you not to be vague in your answers, but I'm asking for help it would be a little rude, but yeah... assume I know little to nothing about this topic
    Blaming your teacher isn't going to help you at all. No matter how how bad your teacher is, you surely should be able to look up the definition of "linearly independent". For functions it is that the only way you can have a_1f_1(x)+ a_2f_2(x)+ \cdot\cdot\cdot+ a_nf_n(x)= 0, for all x, is to have a_1= a_2= \cdot\cdot\cdot= a_n= 0.

    a) If f(x)= sin(2x)+ e^x, what are f'(x), f"(x), and f'''(x)? Suppose a_1f(x)+ a_2f'(x)+ a_3f"(x)+ a_4f'''(x)= 0 for all x. Taking four different values for x will give you four linear equations to solve for a_1, a_2, a_3, and a_4.

    b) If f(x)= x^3+ 1, what are f^2 and f^3. Suppose a_1f(x)+ a_2f^2(x)+ a_3f^3(x)= 0 for all x. Taking three different values for x will give you three linear equations to solve.

    c) I assume that by " cos(x)^2 you mean cos^2(x), not cos(x^2).

    Again, suppose you have a_1cos(2x)+ a_2sin^2(x)+ a_3cos^2(x)= 0. Taking three different values for x will give you three linear equations to solve.
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  3. #3
    Senior Member chella182's Avatar
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    Quote Originally Posted by HallsofIvy View Post
    Blaming your teacher isn't going to help you at all. No matter how how bad your teacher is, you surely should be able to look up the definition of "linearly independent".
    Ok. Thanks for the help and all, I do kind of understand what you're saying which should help with the work, but you are working the rather rude and risky assumption that I haven't tried to look things up for myself. I assure you this website is always my last resort.
    Last edited by chella182; December 11th 2009 at 07:11 AM.
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