Lesson plan
Null vector
In mathematics, given a vector space X with an associated quadratic form q, written , a null vector or isotropic vector is a non-zero element x of X for which . In the theory of real bilinear forms, definite quadratic forms and isotropic quadratic forms are distinct. They are distinguished in that only for the latter does there exist a nonzero null vector. A quadratic space which has a null vector is called a pseudo-Euclidean space. The term isotropic vector v when q(v) = 0 has been used in quadratic spaces, and anisotropic space for a quadratic space without null vectors. A pseudo-Euclidean vector space may be decomposed (non-uniquely) into orthogonal subspaces A and B, , where q is positive-definite on A and negative-definite on B. The null cone, or isotropic cone, of X consists of the union of balanced spheres: \bigcup_{r \geq 0} \{x = a + b : q(a) = -q(b) = r, \ \ a \in A, b \in B \}. The null cone is also the union of the isotropic lines through the origin.
Article summary
In mathematics, given a vector space X with an associated quadratic form q, written , a null vector or isotropic vector is a non-zero element x of X for which . In the theory of real bilinear forms, definite quadratic forms and isotropic quadratic forms are distinct. They are distinguished in that only for the latter does there exist a nonzero null vector. A quadratic space which has a null vector is called a pseudo-Euclidean space. The term isotropic vector v when q(v) = 0 has been used in quadratic spaces, and anisotropic space for a quadratic space without null vectors. A pseudo-Euclidean vector space may be decomposed (non-uniquely) into orthogonal subspaces A and B, , where q is positive-definite on A and negative-definite on B. The null cone, or isotropic cone, of X consists of the union of balanced spheres: \bigcup_{r \geq 0} \{x = a + b : q(a) = -q(b) = r, \ \ a \in A, b \in B \}. The null cone is also the union of the isotropic lines through the origin.
Key claims
No claims recorded.
Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
Bibliography
No sources recorded.
Key claims and bibliography are generated from the same tracked claims and sources as the article page.