Coordinate space

Description

Let $F$ be a field and let $n \ge 0$ be an integer. The coordinate space $F^n$ is the set of $n$-tuples $(x_1, \dots, x_n)$ with $x_i \in F$. It is a vector space over $F$ with the operations $$(x_1, \dots, x_n) + (y_1, \dots, y_n) = (x_1 + y_1, \dots, x_n + y_n), \qquad a(x_1, \dots, x_n) = (ax_1, \dots, ax_n).$$ Its zero vector is $(0, \dots, 0)$.

Remarks

The space $F^n$ has dimension $n$, with the standard basis $e_1, \dots, e_n$. More generally, for a set $I$, the set $F^I$ of all functions $I \to F$ is a vector space with pointwise operations. Each $n$-dimensional vector space over $F$ is isomorphic to $F^n$. The real spaces $\R^2$ and $\R^3$ model the plane and ordinary space.

Explanations

L1 human

The coordinate space $F^n$ is the space of lists of $n$ numbers. We add two lists position by position, and we scale a list by scaling each entry. Every finite-dimensional vector space looks like one of these spaces after we choose a basis.