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Homework Help: Linear Algebra: Vector Spaces; Dependent/Independent

  1. Nov 15, 2011 #1
    Decide if the indicated set of functions are independent or dependent, and prove your answer.

    [tex]
    \left\{cos^2(x),sin^2(x),sin(2x)\right\}
    [/tex]

    This linear algebra course is killing me. It's much more abstract than I thought it would be. I realize this problem isn't exactly that, but I am so overwhelmingly frustrated with this class.

    Dependence/Indepence is determined if one of the vectors is/is not the zero vector. But for functions how do I interpret this? Do I set it up like an equation and set it to the zero vector? AAAHHHH! So. Completely. Lost. I've read this section twice now and I feel I have nothing to show for it.
     
  2. jcsd
  3. Nov 15, 2011 #2
    You need to determine whether:

    asin2(x) + bcos2(x) +csin(2x) = 0

    has any solution except a=b=c=0.

    Basically can you write any of the three as linear cominqtions of the others. Do you know any realtions between sin2, cos2 and sin(2x)?
     
  4. Nov 15, 2011 #3
    I realize that I can't do that, but how do I prove it? I was thinking about constructing a matrix but that seems ridiculous.
     
  5. Nov 15, 2011 #4
    span(u,v) = span(u+v,u-v)

    You can replace sin^2 and cos^2 with sin^2 + cos^2 and sin^2 -cos^2. Can you prove the resulting 3 functions are independent.
     
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