(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I'm having a tough time figuring out just how to get the orthogonal complement of a space. The provlem gives two separate spaces:

1) span{(1,0,i,1),(0,1,1,-i)},

2) All constant functions in V over the interval [a,b]

2. Relevant equations

I know that for a subspace W of an inner product space, the orthogonal complement is defined as:

W_perp = {vectors in v[itex]\in[/itex]V: <v,w> = 0 for all w [itex]\in[/itex] W}

<v,w> is the standard dot product between two vectors;

In the case of constant functions, the dot product is [itex]\int[/itex]f(x)g(x)dx;

3. The attempt at a solution

1) I tried putting it in matrix form:

[1 0 i 1]

[0 1 1 -i]

but don't know how to row reduce with complex variables. I actually don' think the matrix needs to be simplified more than it is, but still don't know how to plug in to get two orthogonal vectors (I would like the result to be an orthogonal set)

2) I don't even know where to start ... the book doesn't cover inner products of functions much , let alone how to find the orthogonal complement of them.

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# Orthogonal Complements of complex and continuous function subspaces

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