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Description

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

  1. u⃗+v⃗∈V\vec{u} + \vec{v} \in V

  2. cu⃗∈Vc\vec{u} \in V

  3. u⃗+v⃗=v⃗+u⃗\vec{u} + \vec{v} = \vec{v} + \vec{u}

  4. (u⃗+v⃗)+z⃗=u⃗+(v⃗+z⃗)(\vec{u} + \vec{v}) + \vec{z} = \vec{u} + (\vec{v} + \vec{z})

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

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

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

  8. (c+d)u⃗=cu⃗+du⃗(c + d)\vec{u} = c\vec{u} + d\vec{u}

  9. c(u⃗+v⃗)=cu⃗+cv⃗c(\vec{u} + \vec{v}) = c\vec{u} + c\vec{v}

  10. 1u⃗=u⃗1\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.

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Definition of a subspace

A subset WW of a vector space VV is a subspace of VV if the following two conditions are satisfied for any u⃗,v⃗∈W\vec{u}, \vec{v} \in W and c∈Rc \in \mathbb{R}.

  1. u⃗+v⃗∈W\vec{u} + \vec{v} \in W

  2. cu⃗∈Wc\vec{u} \in W

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

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