mathbbR4

R4 refers to the four-dimensional real vector space. It is the set of all ordered quadruples of real numbers, which can be written as (x1, x2, x3, x4) where x1, x2, x3, and x4 are real numbers. The operations of vector addition and scalar multiplication are defined component-wise. For example, given two vectors u = (u1, u2, u3, u4) and v = (v1, v2, v3, v4) in R4, their sum is u + v = (u1+v1, u2+v2, u3+v3, u4+v4). Similarly, for a scalar c in R, the scalar multiplication is c * u = (c*u1, c*u2, c*u3, c*u4).

R4 is a fundamental object in linear algebra and has various applications in physics and mathematics. It is a vector space over the field of real numbers, meaning it satisfies the axioms of a vector space. The standard basis for R4 consists of four vectors: e1 = (1, 0, 0, 0), e2 = (0, 1, 0, 0), e3 = (0, 0, 1, 0), and e4 = (0, 0, 0, 1). Any vector in R4 can be expressed as a linear combination of these basis vectors. Geometrically, R4 can be visualized as a space with four independent directions, though direct visualization beyond three dimensions is challenging for humans. It serves as a foundation for understanding higher-dimensional spaces.