Introduction to Vector Spaces

A vector space (over $\mathbb {R}$) consists of a set $V$ along with two operations "$+$" and "$\cdot$" subject to these conditions.

1. For any ${\vec {v}},{\vec {w}}\in V:{\vec {v}}+{\vec {w}}\in V$.
2. For any ${\vec {v}},{\vec {w}}\in V:{\vec {v}}+{\vec {w}}={\vec {w}}+{\vec {v}}$.
3. For any ${\vec {u}},{\vec {v}},{\vec {w}}\in V:({\vec {v}}+{\vec {w}})+{\vec {u}}={\vec {v}}+({\vec {w}}+{\vec {u}})$.
4. There is a zero vector ${\vec {0}}\in V$ such that ${\vec {v}}+{\vec {0}}={\vec {v}}$ for all ${\vec {v}}\in V$.
5. Each ${\vec {v}}\in V$ has an additive inverse ${\vec {w}}\in V$ such that ${\vec {w}}+{\vec {v}}={\vec {0}}$.
6. If $r$ is a scalar, that is, a member of $\mathbb {R}$ and ${\vec {v}}\in V$ then the scalar multiple $r\cdot {\vec {v}}$ is in $V$.
7. If $r,s\in \mathbb {R}$ and ${\vec {v}}\in V$ then $(r+s)\cdot {\vec {v}}=r\cdot {\vec {v}}+s\cdot {\vec {v}}$.
8. If $r\in \mathbb {R}$ and ${\vec {v}},{\vec {w}}\in V$ , then $r\cdot ({\vec {v}}+{\vec {w}})=r\cdot {\vec {v}}+r\cdot {\vec {w}}$.
9. If $r,s\in \mathbb {R}$ and ${\vec {v}}\in V$ , then $(rs)\cdot {\vec {v}}=r\cdot (s\cdot {\vec {v}})$.
10. For any ${\vec {v}}\in V$ , $1\cdot {\vec {v}}={\vec {v}}$.

Remark: Because it involves two kinds of addition and two kinds of multiplication, that definition may seem confused. For instance, in condition 7 "$(r+s)\cdot {\vec {v}}=r\cdot {\vec {v}}+s\cdot {\vec {v}}$", the first "$+$" is the real number addition operator while the "$+$" to the right of the equals sign represents vector addition in the structure $V$ . These expressions aren't ambiguous because, e.g., $r$ and $s$ are real numbers so "$r+s$" can only mean real number addition.