Orthogonal complements | Alternate coordinate systems (bases) | Linear Algebra | Khan Academy

TL;DR

The orthogonal complement is the set of vectors that are orthogonal to all vectors in a given subspace.

Transcript

Say I've got a subspace V. So V is some subspace, maybe of Rn. I'm going to define the orthogonal complement of V, let me write that down, orthogonal complement of V is the set. And, this is shorthand notation right here, would be the orthogonal complement of V. So we write this little orthogonal notation as a superscript on V. And you can pronounc... Read More

Key Insights

• 😫 The orthogonal complement is the set of vectors that are orthogonal to all vectors in a given subspace.
• ❓ The orthogonal complement is a subspace if it satisfies certain conditions.
• 👾 The null space of a matrix is equal to the orthogonal complement of the row space.

Questions & Answers

Q: What is the orthogonal complement of a subspace?

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

Q: How do you determine if the orthogonal complement is a subspace?

The orthogonal complement is a subspace if it satisfies the conditions of closure under addition, closure under scalar multiplication, and contains the zero vector.

Q: What is the relationship between the null space of a matrix and the row space?

The null space of a matrix A is equal to the orthogonal complement of the row space of A. This means that any vector in the null space is orthogonal to every vector in the row space.

Q: How is the left null space related to the column space of a matrix?

The left null space of a matrix A is equal to the orthogonal complement of the column space of A. This means that any vector in the left null space is orthogonal to every vector in the column space.

Summary & Key Takeaways

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

• The orthogonal complement is a subspace if it satisfies the conditions of closure under addition, scalar multiplication, and contains the zero vector.

• The null space of a matrix A is equal to the orthogonal complement of the row space of A, and the left null space is equal to the orthogonal complement of the column space of A.