- #1
Mdhiggenz
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Homework Statement
Hello, I took my quiz today, and had to find a basis for an orthogonal compliment,
would it be incorrect to not factor out the alphas and betas?
An orthogonal compliment is the subspace of a vector space that is perpendicular to a given subspace. In other words, it is the set of all vectors that are orthogonal (or perpendicular) to every vector in the given subspace.
The orthogonal compliment of a subspace can be calculated by finding the basis of the subspace and then using the Gram-Schmidt process to find a basis for the orthogonal compliment. Another method is to use the null space of the transpose of the matrix representation of the subspace.
The orthogonal compliment is an important concept in linear algebra because it allows us to decompose a vector space into two perpendicular subspaces, which can make solving systems of linear equations and performing other operations easier. The orthogonal compliment is also used in applications such as data compression and dimensionality reduction.
Yes, it is possible for the orthogonal compliment of a subspace to be empty. This can occur if the subspace is the entire vector space or if the subspace does not have a basis consisting of orthogonal vectors.
The concept of orthogonality is closely related to the orthogonal compliment. Two vectors are orthogonal if their dot product is equal to zero, and the orthogonal compliment is the subspace of all vectors that are orthogonal to a given subspace. In other words, the orthogonal compliment is the set of all vectors that are orthogonal to every vector in the given subspace.