Real Space

The real space, denoted by ${\mathbb{R}}^n$, is the set of all ordered $n$-tuples of real numbers. Such tuples are usually written as Vectors.

The most common real spaces are:

• $\mathbb{R}$, which contains all real numbers
• ${\mathbb{R}}^2$, a 2D space which encompasses the Cartesian Plane, and is most commonly used for graphing
• ${\mathbb{R}}^3$, a 3D Real Space used to graph 3D functions

Definition §

${\mathbb{R}}^n$ is defined to be the set of all ordered $n$-tuples of real numbers written as row vectors:

$$\vec{v}=(v_1,v_2,v_3,\dots v_n),v_1,v_2,..v_n\in \mathbb{R}$$

Assume we have two vectors in the real space, $\vec{u}$ and $\vec{v}$. Then addition is defined by:

$$\vec{v}+\vec{u}=(v_1,v_2,v_3\dots v_n)+(u_1,u_2,u_3,\dots u_n)=(v_1+u_1,v_2+u_2,\dots v_n+u_n)$$

If we let $k\in \mathbb{R}$, scalar multiplication is define by:

$$k\vec{v}=k(v_1,v_2,v_3\dots v_n)=(k{v}_1,k{v}_2,k{v}_3\dots k{v}_n)$$