The basis of orthogonal complement is a set of vectors that are orthogonal (perpendicular) to a given vector space. This means that the dot product of any two vectors in the basis is equal to zero. The basis of orthogonal complement is also known as the basis of the null space.
To find the basis of orthogonal complement, you can use the Gram-Schmidt process. This involves taking a basis for the vector space and applying the process to each vector in the basis. This will result in a set of orthogonal vectors that span the same space as the original basis. These orthogonal vectors form the basis of the orthogonal complement.
The orthogonal complement of a vector space is a subspace of the original vector space. This means that the basis of orthogonal complement and the original vector space share the same zero vector. Additionally, any vector in the basis of orthogonal complement is orthogonal to all vectors in the original vector space.
The concept of orthogonal complement is important in linear algebra because it allows us to define the concept of perpendicularity in vector spaces. This is useful in many applications, including in geometry, physics, and engineering. Additionally, the basis of orthogonal complement can help us find solutions to systems of linear equations and understand the structure of vector spaces.
Yes, the basis of orthogonal complement can be empty. This occurs when the original vector space is the zero vector space, meaning that it only contains the zero vector. In this case, there are no vectors that are orthogonal to all vectors in the zero vector space, so the basis of orthogonal complement is empty.