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Math Help - Determine wheter a subset is a subspace.

  1. #1
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    Determine wheter a subset is a subspace.

    Hi, I've done some exercises to determine wheter a subset is a subspace, but I don't know how to do the ones with functions, like the following:

    Determine wheter the following subset is a subspace of the vector space of all the functions f(x), x R.

    G={acosx+bsinx: a, b R}

    So, I have to check that G satisfies the additive identity, the closed under addition and closed under escalar properties.

    I think the first condition is satisfied, right? How do I check if the other two are satisfied?
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  2. #2
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    Re: Determine wheter a subset is a subspace.

    Hey Cesc1.

    Hint: What is the definition of addition and scalar multiplication in the vector space? (Also what is the exact definition of a vector in the space)?
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  3. #3
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    Re: Determine wheter a subset is a subspace.

    Remember, what are the elements of G? Are they real numbers? Are they functions?

    How can you add two functions together? How can you multiply a function by a scalar?
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  4. #4
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    Re: Determine wheter a subset is a subspace.

    Quote Originally Posted by chiro View Post
    Hey Cesc1.

    Hint: What is the definition of addition and scalar multiplication in the vector space? (Also what is the exact definition of a vector in the space)?
    A vector space V over a field F is a set V equipped with an operation called (vector) addition, which takes vectors u and v and produces another vector u + v . There is also an operation called scalar multiplication, which takes an element a in F and a vector u in V and produces a vector au in V.

    So, if I take vectors u and v from G, such that u= a1cosx + b1sinx, and v= a2cosx + b2sinx, their sum u+v= (a1cosx+b1sinx)+(a2cosx+b2sinx) can be expressed as (a1+a2)cosx + (b1+b2)sinx, so the closed under addition property is satisfied.

    Also, if I have u= acosx + bsinx, and some number c from R, then cu= c(acosx + bsinx)= (ac)cosx + bc(sinx), which is also in G. So, I think that G is indeed a subspace. Is that right?
    Last edited by Cesc1; August 18th 2013 at 11:41 AM.
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  5. #5
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    Re: Determine wheter a subset is a subspace.

    Yes, that is correct!
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