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Linearly dependent or independent functions?

  1. Aug 31, 2007 #1
    Are the functions y=0 and y=sinh(pi*x) linearly dependent or linearly independent on the intercal x>0?

    I'm not sure what I'm supposed to do here, but I try to divide them:

    0/sinh(pi*x). This is certainly 0, since sinh (pi*x) is positive when x>0. Since 0 is a constant, the functions must be linearly dependent.

    Am I right?
  2. jcsd
  3. Aug 31, 2007 #2
    Any set that contains the 0 vector is linearly dependent.

    In this case, the zero function is the zero vector of the space of functions defined on [o,infinity).

    This is because any vector written as a linear combination of linearly independent vectors should have a unique linear combination... however, with the 0 vector included, you can come up with as many linear combinations as you want.

    Hope this helps
  4. Aug 31, 2007 #3
    Thanks. I haven't learnt that much about matrixes yet. But the way I reasoned is plausible as well?
  5. Aug 31, 2007 #4
    It really depends on what type of class this is for.

    Is this a linear algebra class?
  6. Aug 31, 2007 #5
    Yes. This chapter is called "Second order linear ODEs".

    Another problem involves the functions ln x and ln(x^2). Then I wrote ln(x^2) as 2ln x, and since ln x/2ln x = 1/2, the functions are lineary dependent.
    Last edited: Aug 31, 2007
  7. Aug 31, 2007 #6
    Ahhh, ok.

    I would probably just point out that the functions are linearly dependent because one can be written as a linear combination of the other.

    Let f(x) = 0
    Let y(x) = sin(pi*x)

    f(x) = 0 = 0 * sin(pi*x) = 0 * g(x)

    Since f(x) is written here as a constant times g(x), you have linearly dependent functions.
  8. Sep 1, 2007 #7


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    I was a little puzzled why you divided them. A set of vectors is "independent" if and only if the only linear combination,
    [tex]\alpha_1 v_1+ \alpha_2 v_2+ \cdot\cdot\cdot+ \alpha_n v_n= 0[/tex]
    must have all the [/alpha]s equal to 0.
    Of course, for just two vectors (or functions) that is
    [tex]\alpha_1 f+ \alpha_2 g= 0[/tex]
    must have [itex]\alpha_1= \alpha_2= 0[/itex]. If that's not true, if the functions are dependent, then
    [tex]\frac{f}{g}= -\frac{\alpha_1}{\alpha_2}[/tex]
    exists and is non 0. So your method is correct.
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