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I have a doubt about a problem involving inner product spaces. The exercise is:

Given the subspace generated by the vectors $ (1,1,1) $ and $ (1,-1,0) $, find the orthogonal subspace and give a basis.

Now, what I understood of this problem is that I have to find the orthogonal complement. But they don´t give me any inner product to work with. Does the orthogonal complement vary with the product? Which one should I use?

I haven´t seen much of inner product, just the basic, but I should be able to solve this. Thank you! If you do not understand something, please ask me.

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The orthogonal component depends on the inner product. However, typically the inner product that is chosen (assuming you are dealing with a vector space over real numbers) is the following one: $$\langle(x_1,x_2,x_3), (y_1,y_2,y_3) \rangle = x_1y_1 + x_2y_2 + x_3y_3$$

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  • $\begingroup$ Thanks! That's all i needed to know. So, if they don't clarify which inner product they are using, I just go with the typical one? $\endgroup$
    – Lessa121
    Commented Dec 8, 2013 at 0:02
  • $\begingroup$ @LunaSage Yes ${}{}$ $\endgroup$
    – user17762
    Commented Dec 8, 2013 at 0:02
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    $\begingroup$ @LunaSage "The typical one" being called the Euclidean scalar (or inner) product, in case you need to search for it on the internet or in the literature. $\endgroup$ Commented Dec 8, 2013 at 0:11

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