Question

What are the basic properties of vectors under the operations of vector addition and scalar multiplication?

Answer 1

For any vectors $u, v, w$ in $\mathbf{R}^{n}$ and any scalars $k, k^{\prime}$ in $\mathbf{R}$,

(i) $(u+v)+w=u+(v+w)$,
(ii) $u+0=u$,
(iii) $u+(-u)=0$,
(iv) $u+v=v+u$,

(v) $k(u+v)=k u+k v$,
(vi) $\left(k+k^{\prime}\right) u=k u+k^{\prime} u$
(vii) $\quad\left(\mathrm{kk}^{\prime}\right) \mathrm{u}=\mathrm{k}\left(\mathrm{k}^{\prime} \mathrm{u}\right)$,
(viii) $1 u=u$.