Real vector space

A real vector space is a vector space $V$ where

$$V \in \mathbb{R}^n$$

It can be written as a set:

$$V=\{(x_1,x_2\cdots x_n) | \forall k \in \{1,\cdots ,n\}, x_k \in \mathbb {R} \}$$

Test conditions

A set of vectors $G=\{\vec g_1, \cdot, \vec g_n\}$ is a real vector space if:

• $\vec 0 \in G$, and
• $g_1,\cdots,g_n \in \mathbb{R}^n$, and
• $\sum^{n}_{i=1}\alpha_i g_i \in G$.

Examples

1. Is $G=\{(x,1,y) | x,y \in \mathbb{R}\}$ a vector space?

Solution:

Since $\vec 0=(0,0,0) \notin G$, $G$ is not a vector space.