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Math Help - Sum of Linear Spaces

  1. #1
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    Sum of Linear Spaces

    Hi. Glad the site is running again. By the way, where is the latex add-on?

    Can someone please explain the differences between

    1. PQ, P and Q are linear spaces

    2. P + Q

    3. Sp(PQ)

    Are 2 and 3 the same?
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  2. #2
    A Plied Mathematician
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    P union Q will equal the set of all vectors coming from either P or Q.

    P + Q will equal the set of all vectors that look like p + q, where p in P and q in Q.

    Sp ( P union Q ) will equal the set of all linear combinations of vectors coming from P union Q.

    1 and 2 are not necessarily the same, I wouldn't think: the zero vector of one space need not be the zero vector of another space, although the zero vector of any given space is unique.

    That's all I have time for right now.
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  3. #3
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    well i would think that 2 and 3 are not the same as P + Q is not all linear combinations as it does not include scalar multiplication, where as 3 does?
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  4. #4
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    Quote Originally Posted by hmmmm View Post
    well i would think that 2 and 3 are not the same as P + Q is not all linear combinations as it does not include scalar multiplication, where as 3 does?
    Except that the underlying spaces are linear, which does include scalar multiplication.
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  5. #5
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    ah sorry about that you are completely correct i apologise for that, really was not thinking very thourghly
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