Isotropic quadratic form in the context of "Linear subspace"

In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises by either replacing the parallel postulate with an alternative, or consideration of quadratic forms other than the definite quadratic forms associated with metric geometry. In the former case, one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries. When isotropic quadratic forms are admitted, then there are affine planes associated with the planar algebras, which give rise to kinematic geometries that have also been called non-Euclidean geometry.

In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, $${\displaystyle 4x^{2}+2xy-3y^{2}}$$

is a quadratic form in the variables x and y. The coefficients usually belong to a fixed field K, such as the real or complex numbers, and one speaks of a quadratic form over K. Over the reals, a quadratic form is said to be definite if it takes the value zero only when all its variables are simultaneously zero; otherwise it is isotropic.

In mathematics and theoretical physics, a pseudo-Euclidean space of signature (k, n-k) is a finite-dimensional real n-space together with a non-degenerate quadratic form q. Such a quadratic form can, given a suitable choice of basis (e₁, …, e_n), be applied to a vector x = x₁e₁ + ⋯ + x_ne_n, giving$${\displaystyle q(x)=\left(x_{1}^{2}+\dots +x_{k}^{2}\right)-\left(x_{k+1}^{2}+\dots +x_{n}^{2}\right)}$$ which is called the scalar square of the vector x.

For Euclidean spaces, k = n, implying that the quadratic form is positive-definite. When 0 < k < n, then q is an isotropic quadratic form. Note that if 1 ≤ i ≤ k < j ≤ n, then q(e_i + e_j) = 0, so that e_i + e_j is a null vector. In a pseudo-Euclidean space with k < n, unlike in a Euclidean space, there exist vectors with negative scalar square.