(i)Prove that if{v1,v2}is linearly dependent, then are multiple of each other, that is, there exists a constant c such that v1 = c v2 or v2=cv1. (ii)P

Recall:Linear Dependence:A finite set of vectors ${\displaystyle \{a_1,a_2,\dots ,a_n}$ pf a vector space V over a field F is said to be linearly dependent in V if there exist scalars ${\displaystyle c1,c2,\dots c_n}$, not all zero, in F such that
${\displaystyle c_1{a}_1+c_2{a}_2+\dots +c_n{a}_n=0}$
θ is the zero element of the vector space V.
(i)Given${\displaystyle \{v_1,v_2\}}$is linearly independent i.e ${\displaystyle v_1,v_2}$ are linearly dependent. Let 0 be the zero element. So there exist scalars ${\displaystyle c_1,c_2}$ such that
${\displaystyle c_1{v}_1+c_2{v}_2=0}$,
if any one of the scalar is 0,then the other scalar also becomess 0(since the vectors are non zero).
So for linear dependency both the scalars ${\displaystyle c_1,c_2}$ have to be non-zero.Therefore,
${\displaystyle c_1{v}_1+c_2{v}_2=0}$
${\displaystyle \Rightarrow c_1{v}_1=-c_2{v}_2}$
${\displaystyle \Rightarrow v_1=(-\frac{c_2}{c_1})v_2}$
${\displaystyle \Rightarrow v_1=c{v}_2}$
Similarly,
${\displaystyle c_1{v}_1+c_2{v}_2=0}$
${\displaystyle \Rightarrow c_2{v}_2=-c_1{v}_1}$
${\displaystyle \Rightarrow v_2=(-\frac{c_1}{c_2})v_1}$
${\displaystyle \Rightarrow v_2=c{v}_1}$.
So it is proved that if ${\displaystyle \{v_1,v_2\}}$ is linearly dependent, then there exist a constant c such that either ${\displaystyle v_1=c{v}_2\text{or}{v}_2=c{v}_1}$.
(ii)Let there exist a constant c not equal to 0 such that either ${\displaystyle v_1=c{v}_2(\text{or}{v}_2=c{v}_1)}$.
Then it can be written as,
${\displaystyle v_1=c{v}_2}$
${\displaystyle \Rightarrow v_1-c{v}_2=0}$
Hence by the definition of linear dependence we can say that ${\displaystyle v_1,v_2}$ are linearly dependent i.e.
${\displaystyle \{v_1,v_2\}}$ is linearly dependent