Chapter 4.1, Problem 61E

Illustrate properties $1-10$ of Theorem $4.2$ for $u=\left(2,-1,3,6\right)$, $v=\left(1,4,0,1\right)$, $w=\left(3,0,2,0\right)$, $c=5$, and $d=-2$.

THEOREM $4.2$Properties of Vector Addition and Scalar Multiplication in $R^n$.

Let $u,v$, and $w$ be vectors in $R^n$, and let $c$ and $d$ be scalars.

1. $u+v$ is vector in $R^n$.	Closure under addition
2. $u+v=v+u$	Commutative property of addition
3. $\left(u+v\right)+w=u+\left(v+w\right)$	Associative property of addition
4. $u+0=u$	Additive identity property
5. $u+\left(-u\right)=0$	Additive inverse property
6. $cu$ is a vector in $R^n$.	Closure under scalar multiplication
7. $c\left(u+v\right)=cu+cv$	Distributive property
8. $\left(c+d\right)u=cu+du$	Distributive property
9. $c\left(du\right)=\left(cd\right)u$	Associative property of multiplication
10. $1\left(u\right)=u$	Multiplicative identity property