Exercise 1.2.2.
1.2.2.a
Solution.
Suppose \(\uu+\vv=\uu+\ww\text{.}\) By adding \(-\uu\) on the left of each side, we obtain:
\begin{align*}
-\uu+(\uu+\vv) \amp =-\uu+(\uu+\ww)\\
(-\uu+\uu)+\vv \amp =(-\uu+\uu)+\ww \quad \text{ by A3}\\
\zer+\vv \amp =\zer+\ww \quad \text{ by A5}\\
\vv \amp =\ww \quad \text{ by A4}\text{,}
\end{align*}
which is what we needed to show.
1.2.2.b
Solution.
We have \(c\zer = c(\zer+\zer) = c\zer +c\zer\text{,}\) by A4 and S2, respectively. Adding \(-c\zer\) to both sides gives us
\begin{equation*}
-c\zer+c\zer = -c\zer+(c\zer+c\zer)\text{.}
\end{equation*}
Using associativity (A3), this becomes
\begin{equation*}
-c\zer+c\zer = (-c\zer+c\zer)+c\zer\text{,}
\end{equation*}
and since \(-c\zer+c\zer=\zer\) by A5, we get \(\zer =\zer+c\zer\text{.}\) Finally, we apply A4 on the right hand side to get \(\zer=c\zer\text{,}\) as required.
1.2.2.c
Solution.
Suppose there are two vectors \(\zer_1,\zer_2\) that act as additive identities. Then
\begin{align*}
\zer_1 \amp = \zer_1+\zer_2 \quad \text{ since } \vv+\zer_2=\vv \text{ for any } \vv\\
\amp =\zer_2+\zer_1 \quad \text{ by axiom A2}\\
\amp \zer_2 \quad \text{ since } \vv+\zer_1=\vv \text{ for any } \vv
\end{align*}
So any two vectors satisfying the property in A4 must, in fact, be the same.
1.2.2.d
Solution.
Let \(\vv\in V\text{,}\) and suppose there are vectors \(\ww_1,\ww_2\in V\) such that \(\vv+\ww_1=\zer\) and \(\vv+\ww_2=\zer\text{.}\) Then
\begin{align*}
\ww_1 \amp = \ww_1+\zer \quad \text{ by A4}\\
\amp = \ww_1+(\vv+\ww_2) \quad \text{ by assumption}\\
\amp = (\ww_1+\vv)+\ww_2 \quad \text{ by A3}\\
\amp = (\vv+\ww_1)+\ww_2 \quad \text{ by A2}\\
\amp = \zer+\ww_2 \quad \text{ by assumption}\\
\amp \ww_2 \quad \text{ by A4}\text{.}
\end{align*}