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Math Help - Sum of sets of vectors (or whatever it's called)

  1. #1
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    Sum of sets of vectors (or whatever it's called)

    S+T = {s+t | s \in S, t \in T}

    Can someone please explain this property better?

    Let's say S = {(1,0),(0,1)} and T = {(2,1),(1,3)}

    What would be S+T?

    I would just assume that S+T would be each vector from S plus each vector from T:

    S+T = {(3,1),(2,3),(2,2),(1,4)}

    But obviously I'm wrong since S+T is a subspace and what I have is clearly not a subspace. How do you get the 0 vector from S+T?

    Thanksss.

    ***Correction - S+T is a subspace only if S and T are both subspaces, and that would explain the zero vector thing... So is my calculation correct?? And does that mean that in this particular example S+T is not a subspace??
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  2. #2
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    Quote Originally Posted by jayshizwiz View Post
    S+T = {s+t | s \in S, t \in T}

    Can someone please explain this property better?

    Let's say S = {(1,0),(0,1)} and T = {(2,1),(1,3)}

    What would be S+T?

    I would just assume that S+T would be each vector from S plus each vector from T:

    S+T = {(3,1),(2,3),(2,2),(1,4)}
    Yes, that is correct.

    But obviously I'm wrong since S+T is a subspace and what I have is clearly not a subspace. How do you get the 0 vector from S+T?
    Why do you conclude that S+ T is a subspace?

    Thanksss.

    ***Correction - S+T is a subspace only if S and T are both subspaces, and that would explain the zero vector thing... So is my calculation correct?? And does that mean that in this particular example S+T is not a subspace??
    Yes, S and T are not subspaces so S+ T is not necessarily a subspace. And in this case, it isn't.
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  3. #3
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    Why do you conclude that S+ T is a subspace?
    That was simply my original thought. After I posted this, I edited my post with the ***Correction part

    S+T = {s+t | s S, t T}
    So what exactly is this defintion called in English? What is S+T?

    One last thought:

    Let's say S = {(1,0),(0,1)} and T = {(2,1),(1,3)}
    Can I say that (1,1) is also in S+T since it is the sum of the two vector of S? Or, you must choose 1 vector from S and 1 vector from T?
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  4. #4
    Senior Member roninpro's Avatar
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    This is called an internal direct sum.

    Direct sum - Wikipedia, the free encyclopedia
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  5. #5
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    This is called an internal direct sum.
    It is not exactly/always a direct sum:

    U = (a, b, c | a+b+c=0)

    W = (a, b, c | a=c)

    In this case U+W = R^3

    but (0,0,0) = (0,0,0) + (0,0,0)
    but also (0,0,0) = (1,-2,1) + (-1,2,-1)

    So Maybe in general S+T is simply called an internal sum
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