Orthogonal complement question

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In summary, To find the orthogonal complement of a set, take the null space of its transpose. In this problem, the solution space of the homogenous system Ax=0 has three free variables and can be represented by the equations x1 = x1, x2 = (an expression involving some of x1, x2, x3), x3 = x3, x4 = x4.
  • #1
Geekster
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I have the set

[tex]
s = span ( [[0][1][-1][1]]^{T} )
[/tex]

And I need to find the orthogonal complement of the set.

It seems like it should be straight foward, but I'm a bit confused. I know that S is a subspace of R^4, and that there should be three free vairables.

What I did so far is to take the column vector given, and I need to find the null space of its transpose. The three free variables I picked are [tex]x_1= s, x_2=t, x_3=w, x_4=t-w[/tex].

However, x_1=s is throwing me off because its always zero. I guess what I'm really asking is, what exactly is the solution space of the homogenous system,
[tex]
Ax=0
[/tex]
in this problem?

Thanks
 
Last edited:
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  • #2
You row reduce it (actually it is already row reduced) and you get 3 free variables: x1, x3, and x4. You have a pivot for x2. Your equations will be
x1 = x1
x2 = (an expression involving some of x1, x2, x3)
x3 = x3
x4 = x4
 

1. What is the definition of an orthogonal complement?

An orthogonal complement is a subspace of a vector space that contains all vectors that are perpendicular to a given subspace.

2. How is the orthogonal complement of a subspace calculated?

The orthogonal complement of a subspace can be calculated by finding a basis for the subspace and then using the Gram-Schmidt process to find an orthogonal basis. The orthogonal complement is then the subspace spanned by the orthogonal basis.

3. What is the relationship between a subspace and its orthogonal complement?

A subspace and its orthogonal complement are always perpendicular to each other, meaning that their intersection is only the zero vector. In other words, every vector in the subspace is perpendicular to every vector in the orthogonal complement.

4. Can the orthogonal complement of a subspace have a different dimension than the subspace itself?

Yes, the orthogonal complement of a subspace can have a different dimension than the subspace. For example, in a three-dimensional space, a two-dimensional subspace would have a one-dimensional orthogonal complement.

5. What is the significance of the orthogonal complement in linear algebra?

The orthogonal complement plays a crucial role in linear algebra, particularly in the study of vector spaces and linear transformations. It allows for the decomposition of a vector space into two complementary subspaces, making it easier to solve systems of equations and understand the geometry of higher-dimensional spaces.

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