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    In the normed linear space (\mathbb{K}^{3},||.||_2) find the orthogonal complement where Y is the linear subspace spanned by (3,1,-1) and (1,-1,2).

    can someone just check my working please:

    <(a,b,c),(3,1,-1)>=0 so 3a+b-c=0
    <(a,b,c),(1,-1,2)>=0 so a-b+2c=0

    solving gives the complement = {\alpha(-1,7,4): \alpha \in \mathbb{K}}

    I thought this was correct but I don't see where ||.||2 comes into it?
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    (Original post by Big_Sam)
    In the normed linear space (\mathbb{K}^{3},||.||_2) find the orthogonal complement where Y is the linear subspace spanned by (3,1,-1) and (1,-1,2).

    can someone just check my working please:

    <(a,b,c),(3,1,-1)>=0 so 3a+b-c=0
    <(a,b,c),(1,-1,2)>=0 so a-b+2c=0

    solving gives the complement = {\alpha(-1,7,4): \alpha \in \mathbb{K}}

    I thought this was correct but I don't see where ||.||2 comes into it?
    I'm very rusty on this so could be wrong: The fact that it is ||.||2 means that you can derive an inner product on the space, and hence your calculation becomes valid, which is not the case with a different norm.

    Really needs someone more knowledgeable for the definitive answer.
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    (Original post by ghostwalker)
    I'm very rusty on this so could be wrong: The fact that it is ||.||2 means that you can derive an inner product on the space, and hence your calculation becomes valid, which is not the case with a different norm.

    Really needs someone more knowledgeable for the definitive answer.
    Yeah, I'd basically agree with this. ||.||2 is just the standard Euclidean norm on K^3 - this is the norm that gives our usual inner product (multiply the components and add). The reason that this is explicitly stated is that there also exist other, more exotic norms/inner products.
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    (Original post by ljfrugn)
    Yeah, I'd basically agree with this. ||.||2 is just the standard Euclidean norm on K^3 - this is the norm that gives our usual inner product (multiply the components and add). The reason that this is explicitly stated is that there also exist other, more exotic norms/inner products.
    Ah ok I see what you mean. So what I have done is right after all.

    Thanks
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    (Original post by ljfrugn)
    Yeah, I'd basically agree with this. ||.||2 is just the standard Euclidean norm on K^3 - this is the norm that gives our usual inner product (multiply the components and add). The reason that this is explicitly stated is that there also exist other, more exotic norms/inner products.
    More precisely: \| \cdot \|_2 arises from the standard inner product, but any inner product whatsoever induces a norm. Not every norm arises from an inner product, however - indeed, \| \cdot \|_p is induced by an inner product if and only if p = 2.
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    (Original post by Zhen Lin)
    More precisely: \| \cdot \|_2 arises from the standard inner product, but any inner product whatsoever induces a norm. Not every norm arises from an inner product, however - indeed, \| \cdot \|_p is induced by an inner product if and only if p = 2.
    So what everyone is saying is that what I have done is only correct for the ||.||_2 for another norm this calculation would not have worked?
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    There isn't such a thing as an orthogonal complement in a normed space, in general. What you have there is actually an inner product space.
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    (Original post by Zhen Lin)
    More precisely: \| \cdot \|_2 arises from the standard inner product, but any inner product whatsoever induces a norm.
    Well, it depends on how you look at it. A norm gives rise to an inner product (via the polarisation identity) if and only if it satisfies the parallelogram law.
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    That was probably slightly poorly worded - I mean that given a norm, there exists an inner product induced by that norm if and only if the norm satisfies the parallelogram law. This inner product is then given by the polarisation identity.
 
 
 
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