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Math Help - linear independence

  1. #1
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    linear independence

    CAN somebody,please write down the definition for the linear independence of the following functions?

    {  e^x,e^{2x}}
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  2. #2
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    Quote Originally Posted by alexandros View Post
    CAN somebody,please write down the definition for the linear independence of the following functions?

    {  e^x,e^{2x}}
    As in any vector space, the set \{e^x, e^{2x}\} is linearly independent iff whenever there are scalars a,b \in \mathbb{F} (where \mathbb{F} is your underlying field) such that ae^x + be^{2x} =0 we must have a=b=0.

    In this particular case putting x=0 we have a=b and so ae^x(1+e^x)=0 but 0 \neq e^x for all x \in \mathbb{F} (assuming \mathbb{F} = \mathbb{R} or \mathbb{C} ) so a(1+e^x)=0 for all x \in \mathbb{F} and since e^x is not constant we must have b=a=0 and so your set is linearly independent.
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  3. #3
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    Your argument is not quite correct. At x=0 we have a+b=0 i.e. a=-b
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  4. #4
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    Use the Wronskian and you'll find linear indepencence.
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  5. #5
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    Quote Originally Posted by Krizalid View Post
    Use the Wronskian and you'll find linear indepencence.
    I am sorry i am iterested in the definition and not the proof
    thanks
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  6. #6
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    Quote Originally Posted by alexandros View Post
    I am sorry i am iterested in the definition and not the proof
    thanks
    Jose27 gave you the definition:

    Quote Originally Posted by Jose27 View Post
    As in any vector space, the set \{e^x, e^{2x}\} is linearly independent iff whenever there are scalars a,b \in \mathbb{F} (where \mathbb{F} is your underlying field) such that ae^x + be^{2x} =0 we must have a=b=0.
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