Properties of vector spaces

Recall our

A vector space over a field $\mathbb{F}$ is a set $V$ with two operations:

Addition: $+:V \times V \rightarrow V$, and
Scalar multiplication: $\cdot : \mathbb{F} \times V \rightarrow V $

such that the following axioms hold for all $u,v,w \in V$ and all $a,b \in \mathbb{F}$.

Commutativity. $u+v = v+u$
Associativity. $ (u+v)+w = u+(v+w) $ and $ (ab)v = a(b v) $
Additive Identity. Exists $0 \in V$ such that $v + 0 =v $
Additive Inverse. For all $v \in V$ there exists $z \in V$ such that $v+z=0$
Multiplicative Identity. $1v=v$
Distributive Properties. $a (u + v) = au + av$ and $a+b)u = au + bu$

which is an additive abelian group with a field action.

Using this defition, let’s prove a few easy properties of vector spaces.

First of all, we are allowed to “cancel.”

Suppose $V$ is a vector space $u,v,w \in V$, and \begin{equation} u+w = v+w. \label{eq:tocancel} \end{equation} Then $u = v$.

In the following proof, we will be careful to justify each equality with an annotation. Our aim is to help you understand what constitutes a careful proof.

The additive inverse property guarantees $w$ has an inverse "$-1$" in $V$, i.e. \begin{equation} w+(-w) = 0_V. \label{eq:addinv} \end{equation} So, $$ u \stackrel{add. id.}{=} u + 0_V \stackrel{eq. \eqref{eq:addinv}}{=} u + (w = (-w)) $$ $$ \stackrel{assoc.}{=} (u+w) +(-w) \stackrel{eq \eqref{eq:tocancel}}{=} (v+w)+(-w)$$ $$ \stackrel{assoc.}{=} v+ (w + (-w)) \stackrel{\eqref{eq:addinv}}{=} v+0 \stackrel{add. id.}{=} v. $$

Let’s make some observations about this proof so you understand what is expected of you on homeworks, quizzes, and exams.

We used proper English grammar, including sentences and punctuation.
Every time something is claimed to exist, or two things are asserted to be equal, there is an explicit justification such as "by the X property," "by equation Y." This justification can be given in words or by labeling equalities.
Even seemingly obvious equalities such as \begin{equation} (u+v)+w = u+(v+w) \label{eq:just} \end{equation} are justified explicitly.
As we progress in the class, we can take more for granted. On this week's quiz, \eqref{eq:just} must be justified by the associative property. By the fifth week it need not be.
Part of writing a proof is understanding the appropriate level of the proof: what the reader can be assumed to know, and what steps can be skipped.

When in doubt, write it out. I will take off points for insufficient justification, especially in the first third of the class.

Let’s continue and prove some other properties of vector spaces. Two immediate corollaries of the cancellation Proposition above are that

The special $0_V \in V$ vector is unique, and
The additive inverse of any vector is unique.