Orthogonal Complements of complex and continuous function subspaces

In summary, the student is having trouble figuring out how to get the orthogonal complement of a space. The problem is that the book does not cover inner products of functions much, so the second question is not very easy. The student tried to solve the equations for the orthogonal complement of a space using the unknown vectors w=(w1,w2,w3,w4) but did not get the answer correctly.
  • #1
unquantified
2
0

Homework Statement



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]

Homework 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;

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 into 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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  • #2
You doing a pretty good job of ignoring any information in your "Relevant equations" section. If you don't do that the second question should be pretty easy. You can factor a constant function outside of the integral. For the first one you should remember, if you weren't told, that <u,v> for complex vectors involves taking a complex conjugate of one of the vectors. Suppose v=(A,B,C,D). Then what two equations do you have to solve for the four unknowns A, B, C and D?
 
  • #3
So, for the first part, the orthogonal complement would just be all f(x) where by ∫f(x) = 0 over the given interval [a,b]?

As for the second, I'm not quite sure what you mean. If I have those equations, and let x =[x_1, x_2, x_3, x_4] be my unknowns, I have:

[x_1,x_2,x_3,x_4] = x_3[-i, -1, 1, 0] + x_4[-1,i,0,1]

Is that correct? I would then plug in x_3=1 and then x_4=1 to each of those vectors to get my orthogonal complement.
 
  • #4
unquantified said:
So, for the first part, the orthogonal complement would just be all f(x) where by ∫f(x) = 0 over the given interval [a,b]?

As for the second, I'm not quite sure what you mean. If I have those equations, and let x =[x_1, x_2, x_3, x_4] be my unknowns, I have:

[x_1,x_2,x_3,x_4] = x_3[-i, -1, 1, 0] + x_4[-1,i,0,1]

Is that correct? I would then plug in x_3=1 and then x_4=1 to each of those vectors to get my orthogonal complement.

That's the answer to the first one alright. For the second one, take w=(w1,w2,w3,w4) to be your unknown vector in the orthogonal complement. Now since w need to be orthogonal to the span{(1,0,i,1),(0,1,1,-i)}, it has to be orthogonal to v1=(1,0,i,1) and v2=(0,1,1,-i). So you must have <v1,w>=0 and <v2,w>=0. What do those equations look like when you write them out in terms of w1, w2, w3 and w4?
 

1. What is an orthogonal complement?

An orthogonal complement of a subspace is a set of vectors that are perpendicular (orthogonal) to every vector in that subspace. In other words, it is a set of vectors that, when added to the subspace, result in a new subspace that is perpendicular to the original subspace.

2. What is the significance of orthogonal complements in complex and continuous function subspaces?

In complex and continuous function subspaces, orthogonal complements are important because they help us understand the relationship between different subspaces. They also allow us to decompose a given subspace into two smaller subspaces that are orthogonal to each other.

3. How do you find the orthogonal complement of a subspace?

To find the orthogonal complement of a subspace, we first need to find a basis for the subspace. Then, we use the Gram-Schmidt process to find a basis for the orthogonal complement. This involves finding vectors that are perpendicular to the basis vectors of the subspace and normalizing them to create an orthonormal basis for the orthogonal complement.

4. Can the orthogonal complement of a subspace be empty?

Yes, the orthogonal complement of a subspace can be empty. This happens when the subspace itself is the entire vector space, in which case there are no vectors that are perpendicular to every vector in the subspace.

5. How are orthogonal complements related to the concept of orthogonality?

Orthogonal complements are directly related to the concept of orthogonality. In fact, the definition of an orthogonal complement is based on the idea of orthogonality. Two vectors are orthogonal if their dot product is equal to 0, and the orthogonal complement of a subspace is a set of vectors that are orthogonal to every vector in that subspace.

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