Question

Let V be the set R⁺ of positive real number and define V₁ $\bigoplus$ V₂=V₁V₂ and $\delta$ $\bigodot$ V₁=V₁$\delta$ for all V₁V₂$\epsilon$ V and $\delta$ $\epsilon$ R. Then show that v is a vector space over R

Answer 1

A vector space over $\R$ consist of a set $V$ on which is defined of addition associated to elements $v_1\>and\>v_2\>of\>V,\>$ $an \> element \>u\>and\>v\>of \>V$

And an operation of multiplication by scalars,associated to each element $\delta$ of $\R$ and to each element $v\>of\>V$

an element $\delta\>v\>of\>V$

$V$ satisfies the following axioms

$1.\>v_1+v_2=v_2+v_1$

$2.\>(v_1+v_2)+v_3=v_1+(v_2+v_3)$

$3.$ There exists a 0 element$\>0\>of\>V$ such that$\>v+0=v$

$4.$ Given any element $v\>of\>V$ there exists $\>-v\>of\>V$ with the property that

$v+(-v)=0$

$5.$ $(\delta _1+\delta_2)v=\delta_1v+\delta_2v$

$6. \>\delta(v_1+v_2)=\delta\>v_1+\delta\>v_2$

$7.\>\delta_1(\delta_2v)=(\delta_1\delta_2)v$

$8.\>Iv=v$

Where $I$ is the multiplicative identity element of $\R$

Therefore $V$ is a vector space over $\R$