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# 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.