Real Coordinate Space¶

Definition¶

The real coordinate space of dimension $n$ , denoted $\mathbb{R}^{n}$ , is the set of all ordered $n$ -tuples of real numbers.

Interpretation¶

Real coordinate space $\mathbb{R}^n$ is the collection of all ordered lists of $n$ real numbers. Each tuple $\boldsymbol{x}=(x_1,\ldots,x_n)$ is a vector, and its coordinates are the real entries $x_1,\ldots,x_n$ .

Three coordinate axes x1, x2 and x3 with a point x = (x1, x2, x3) and dashed lines projecting it onto the axes.

For $n=3$ $3$ , the figure places the tuple $\boldsymbol{x}=(x_1,x_2,x_3)$ as a point relative to three axes. The dashed lines show how the point is reached: $x_1$ along the first axis, $x_2$ along the second, and then $x_3$ upwards. In dimensions two and three the Cartesian coordinate system gives this picture. For $n\ge4$ there is no picture, but the tuples and the rules for computing with them work in the same way.

Two tuples are equal if and only if they agree in every position, so order matters: $(1,2)$ and $(2,1)$ are different elements of $\mathbb{R}^2$ . With coordinate-wise addition and scalar multiplication, $\mathbb{R}^n$ is the standard example of a real vector space. The $n$ vectors with one coordinate $1$ and all others $0$ form a basis, so $\mathbb{R}^n$ has dimension $n$ .

A tuple can stand for a point, an arrow or simply a list of $n$ numerical values. For example, the values of the $n$ decision variables of a linear program form one point of $\mathbb{R}^n$ . Real coordinate space is the standard setting for doing linear algebra and multivariable calculus.

Key Results¶

With component-wise addition and scalar multiplication, the real coordinate space is a real vector space.

Examples¶

Special cases are called the real line $\mathbb{R}^1$ , the real coordinate plane $\mathbb{R}^2$ , and the real coordinate three-dimensional space $\mathbb{R}^3$ .

Prerequisites¶

Sources¶

Math II: Linear Algebra · Week 2
Math III: Analysis · Lecture 4

Aliases: real n-space Keywords: R^n, n-tuple Symbols: real numbers

Last updated: 20 October 2026 03:02