Orthogonal complement of the orthogonal complement | Linear Algebra | Khan Academy

TL;DR

The video explains the concept of the orthogonal complement and double orthogonal complement of a subspace in linear algebra.

Transcript

Let's say I have some subspace of rn called v. Let me draw it like this. So that it is r n. And I have some subspace of it we'll call v right here. So that is my subspace v. We know that the orthogonal complement v is equal to the set of all of the members of rn. So x is a member of rn. Such that x dot v is equal to 0 for every v that is a member o... Read More

Key Insights

• 🥰 The orthogonal complement of a subspace V consists of vectors that are orthogonal to all vectors in V.
• 😫 The double orthogonal complement of V is the set of vectors orthogonal to all vectors in the orthogonal complement of V.
• 🟰 The double orthogonal complement of V is equal to V, meaning V = (V perp) perp.

Questions & Answers

Q: What is the definition of the orthogonal complement of a subspace?

The orthogonal complement of a subspace V is the set of all vectors in Rn that are orthogonal to every vector in V.

Q: How is the double orthogonal complement of a subspace defined?

The double orthogonal complement of a subspace V is the orthogonal complement of the orthogonal complement of V.

Q: Can vectors in the orthogonal complement of the orthogonal complement of V be represented as a sum of vectors in V and its orthogonal complement?

Yes, any vector x in the orthogonal complement of the orthogonal complement of V can be represented as the sum of a vector in V and a vector in the orthogonal complement of V.

Q: What is the relationship between the double orthogonal complement and the original subspace V?

The double orthogonal complement is equal to the original subspace V, meaning V = (V perp) perp.

Summary & Key Takeaways

• The orthogonal complement of a subspace V is the set of all vectors in Rn that are orthogonal to every vector in V.

• The double orthogonal complement of a subspace V is the orthogonal complement of the orthogonal complement of V.

• It is proven that the double orthogonal complement is equal to the original subspace, meaning V = (V perp) perp.