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Math Help - Linear Subspace problem.

  1. #1
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    Linear Subspace problem.

    I am very sorry if I am bothering you guys a lot but I am trying to understand some of this math and its very hard to do it alone

    So I have this problem that says: Find which of the following sets are linear subspaces of the corresponding linear spaces:

    I am only going to show the first one because maybe if I understand it I can do the rest(I have the answers but I have no idea how it comes out). This is the problem:

    X_{0}=\left \{ (x1,...\: xn)/ x1+...+xn=0 \right.\left.  \right \} \subset \mathbb{R}^{n}, (\mathbb{R}^{n}, \mathbb{R}, \cdot )

    Now I understand to be a linear subspace the set must contain the null vector, be closed under multiplication and addition. My problems begin from the step that I have no idea what (\mathbb{R}^{n} , \mathbb{R}, \cdot ) is.

    Also are these the vectors within (x1,...\: xn) the set X and if so what does this do: / x1+...+xn=0
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  2. #2
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    \mathbb{R} is the set of real numbers.

    \mathbb{R}^n is the set of n-tuples of real numbers.

    I would assume that the notation (\mathbb{R}^n, \mathbb{R}, \cdot ) means that you're looking at the vector space of n-tuples of real numbers over the field of reals where the operation is normal multiplication.

    Now, the null vector is in the subset because 0+\cdots +0 = 0

    If (x_1,\ldots ,x_n), (y_1,\ldots ,y_n) are in the subset, then
    x_1+y_1+\ldots x_n+y_n = x_1+\ldots +x_n+ y_1+\ldots +y_n=0+0=0.
    So (x_1,\ldots ,x_n) + (y_1,\ldots ,y_n) is in the subset.

    Scalar multiplication is similar.

    Yes, it's a subspace.
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  3. #3
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    Ok I think am getting it but whats confusing is that my book talked only about "internal binary operation on V(I guess in this case would be {R}^n), called addition and denoted by +, such that (V,+) is a commutative group". So you can only imagine when they decided to throw in a * how confused I got.

    But am I correct to assume that is the same thing but instead of + its *?
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  4. #4
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    To be honest, that notation is quite confusing. I'm not sure why they chose those 3 things to emphasize (as opposed to addition, for example). I wouldn't worry too much about it though - I think the question is pretty clear. Maybe ask your teacher why this notation is being used.
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  5. #5
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    It is somewhat surprising to lean that mathematical notation is anything but standard. There is a series of probability books all by the same author in which notation varies from book to book. That is why is almost pointless to ask about notation is a forum such as this.
    Sometimes if you give the title and author it is possible that someone may know that particular textbook.

    That said, I think Steve is correct in his guess.
    \left( {\mathbb{R}^n ,\mathbb{R}, \cdot } \right) tells us that the v-space is the set of real n-tuples with vector addition, the scalar field is \mathbb{R} and \cdot tells us the scalar multiplication is that of the real numbers.

    But that is surely explained in the text material.
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