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Math Help - Basis for Subspace

  1. #1
    Member Ranger SVO's Avatar
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    Basis for Subspace

    In the vector space of all real-valued functions, find a
    basis for the subspace spanned by {sin(t); sin(2t); sin(t)cos(t)}

    I can clearly see that sin(2t) can be written as 2*sin(t)cos(t), so that means that 2*sin(t)cos(t) and sin(t)cos(t) are linearly dependent sets.

    Am I right so far?

    So where do I go from here?
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Ranger SVO View Post
    In the vector space of all real-valued functions, find a
    basis for the subspace spanned by {sin(t); sin(2t); sin(t)cos(t)}

    I can clearly see that sin(2t) can be written as 2*sin(t)cos(t), so that means that 2*sin(t)cos(t) and sin(t)cos(t) are linearly dependent sets.

    Am I right so far?
    Yup.

    So where do I go from here?
    Since sin(t) and sin(2t) are linearly independent it looks to me like your basis is {sin(t), sin(2t)} or {sin(t), sin(t)cos(t)}.

    -Dan
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  3. #3
    Member Ranger SVO's Avatar
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    Quote Originally Posted by topsquark View Post
    Since sin(t) and sin(2t) are linearly independent it looks to me like your basis is {sin(t), sin(2t)} or {sin(t), sin(t)cos(t)}.

    -Dan
    I thank you for your response, but can it really be that simple?
    Is it possible to have 2 basis for the same set?
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  4. #4
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Ranger SVO View Post
    I thank you for your response, but can it really be that simple?
    Is it possible to have 2 basis for the same set?
    Of course. Two possible bases for a 2D Euclidean space are the familiar:
    {i, j} <-- Cartesian basis
    {r, theta} <-- Plane polar basis
    (where all of the above are unit vectors.)

    -Dan
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    Quote Originally Posted by topsquark View Post
    Since sin(t) and sin(2t) are linearly independent it looks to me like your basis is {sin(t), sin(2t)} or {sin(t), sin(t)cos(t)}.
    I want to add to topsquarks post.
    Since he does not explain why,
    {sin(t),sin(t)cos(t)}
    Those two a linearly independent (and hence for a basis for its subspace).

    It is because these are not konstant multiples of each other.
    Since,
    sin(t)!=ksin(t)cos(t)

    Using the special theorem for exactly two vectors.
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    Are they really linearly independent??

    are sin(t) and sin(t)*cos(t) really linearly independent???

    because when t=0 both are zero and so is the linear combination and nt their scalar multiples...??
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  7. #7
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    Quote Originally Posted by arpit View Post
    are sin(t) and sin(t)*cos(t) really linearly independent???

    because when t=0 both are zero and so is the linear combination and nt their scalar multiples...??
    What you need to check is that if there exist c_1,c_2 so that c_1 \sin t + c_2 \sin t \cos t = 0 for all t in some interval.
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