Description

A vector space $V$ is defined to be a set of elements that, for any $\vec{u}, \vec{v}, \vec{z} \in V$ and $c, d \in \mathbb{R}$, satisfies the following 10 properties:

1. $\vec{u} + \vec{v} \in V$

2. $c\vec{u} \in V$

3. $\vec{u} + \vec{v} = \vec{v} + \vec{u}$

4. $(\vec{u} + \vec{v}) + \vec{z} = \vec{u} + (\vec{v} + \vec{z})$

5. There is a $\vec{0} \in V$, such that $\vec{u} + \vec{0} = \vec{u}$.

6. There exists a $-\vec{u}$, such that $\vec{u} + (-\vec{u}) = \vec{0}$.

7. $c(d\vec{u}) = (cd)\vec{u}$

8. $(c + d)\vec{u} = c\vec{u} + d\vec{u}$

9. $c(\vec{u} + \vec{v}) = c\vec{u} + c\vec{v}$

10. $1\vec{u} = \vec{u}$

Most of these conditions are obvious, but the most important ones are the no escape properties (properties 1 and 2). In general, you do not need to memorize the 10 properties of vector spaces because we will hardly be dealing with vector spaces as a whole; instead, we will mostly use subspaces.

Definition of a subspace

A subset $W$ of a vector space $V$ is a subspace of $V$ if the following two conditions are satisfied for any $\vec{u}, \vec{v} \in W$ and $c \in \mathbb{R}$.

1. $\vec{u} + \vec{v} \in W$

2. $c\vec{u} \in W$

Consequently, to test if a subset forms a subspace, we need to check whether the 2 above properties are satisfied.