Find A Basis For The Orthogonal Complement

Finding a Basis for the Orthogonal Complement

Finding a basis for the orthogonal complement of a subspace is a fundamental concept in linear algebra with applications in various fields, including machine learning, computer graphics, and quantum mechanics. This article will guide you through the process, explaining the underlying theory and providing practical examples. Understanding orthogonal complements allows you to decompose vector spaces and solve problems involving projections and least squares approximations.

What is an Orthogonal Complement?

Given a subspace W of a vector space V with an inner product (dot product in Rⁿ), the orthogonal complement of W, denoted as W^⊥ (W perp), is the set of all vectors in V that are orthogonal (perpendicular) to every vector in W. In simpler terms, a vector is in W^⊥ if its dot product with every vector in W is zero.

Key Properties of Orthogonal Complements:

• W ∩ W^⊥ = {0}: The only vector common to both W and W^⊥ is the zero vector.
• (W^⊥)^⊥ = W: The orthogonal complement of the orthogonal complement of W is W itself.
• V = W ⊕ W^⊥: V is the direct sum of W and W^⊥; every vector in V can be uniquely expressed as the sum of a vector in W and a vector in W^⊥. This property is crucial for applications like orthogonal projections.

Finding a Basis for the Orthogonal Complement: A Step-by-Step Guide

Let's assume we have a subspace W spanned by a set of linearly independent vectors. To find a basis for W^⊥, we can follow these steps:

1. Represent W using a matrix: If W is spanned by the vectors {v₁, v₂, ..., v_k}, construct a matrix A where each vector v_i is a column.

2. Find the null space of A^T: The orthogonal complement W^⊥ is the null space of the transpose of matrix A, denoted as N(A^T). The null space is the set of all vectors x such that A^Tx = 0.

3. Solve the homogeneous system: Solving the system of linear equations A^Tx = 0 will give you the solutions that form the null space.

4. Determine a basis for the null space: The linearly independent vectors that span the null space form a basis for W^⊥. Techniques like Gaussian elimination can be used to find these linearly independent vectors.

Example:

Let's say W is a subspace of R³ spanned by the vectors v₁ = (1, 0, 1) and v₂ = (0, 1, 1). To find a basis for W^⊥:

1. Matrix A:

   A =  [ 1  0 ]
        [ 0  1 ]
        [ 1  1 ]
   

2. Transpose A^T:

   AT = [ 1  0  1 ]
              [ 0  1  1 ]
   

3. Solve A^Tx = 0: This leads to the system of equations:

   • x₁ + x₃ = 0
   • x₂ + x₃ = 0

4. Find the basis: Solving this system gives x₁ = -x₃ and x₂ = -x₃. Letting x₃ = 1, we get a vector x = (-1, -1, 1). This vector forms a basis for W^⊥. Therefore, the orthogonal complement W^⊥ is spanned by the vector (-1, -1, 1).

Conclusion:

Finding a basis for the orthogonal complement involves understanding the relationship between a subspace and its orthogonal counterpart. By using matrix operations and solving homogeneous systems of equations, we can systematically determine a basis for W^⊥, a powerful tool with significant applications in various mathematical and computational fields. Remember to utilize tools like Gaussian elimination to efficiently solve the homogeneous system and find the linearly independent vectors that form the basis of the orthogonal complement.