mathbbRn

R^n is the set of all n-tuples of real numbers. It is also known as n-dimensional Euclidean space. The elements of R^n are vectors of the form (x1, x2, ..., xn), where each xi is a real number. R^n is a fundamental object in mathematics, particularly in linear algebra, calculus, and analysis.

Vector addition in R^n is performed componentwise. If u = (u1, u2, ..., un) and v = (v1, v2, ..., vn) are two vectors in R^n, then their sum u + v is given by (u1+v1, u2+v2, ..., un+vn). Scalar multiplication is also performed componentwise. If c is a real number and u = (u1, u2, ..., un), then cu = (cu1, cu2, ..., cun).

R^n is a vector space over the field of real numbers. It has a standard basis consisting of n vectors, where each vector has a 1 in one position and 0s elsewhere. For example, in R^3, the standard basis vectors are e1 = (1, 0, 0), e2 = (0, 1, 0), and e3 = (0, 0, 1).

The concept of distance and angles can be defined in R^n using the dot product. The dot product of two vectors u and v is defined as u · v = u1v1 + u2v2 + ... + unvn. The Euclidean norm (or length) of a vector u is ||u|| = sqrt(u · u) = sqrt(u1^2 + u2^2 + ... + un^2). The Euclidean distance between two vectors u and v is ||u - v||.

R^n serves as a generalization of the familiar 2-dimensional plane (R^2) and 3-dimensional space (R^3) to any number of dimensions. This generalization allows for the study of complex systems and phenomena that cannot be easily visualized in lower dimensions.