orthogonal complement

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.

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.

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.

More precisely: $\| \cdot \|_2$ arises from the standard inner product, but any inner product whatsoever induces a norm.