Question 1177931
(==>) Let {{{v[1] = a*v[2]}}} for some real a.

==> {{{v[1] = a*v[2] + 0*v[3]}}} ==> {{{v[1]}}} is a linear combination of {{{v[2]}}} and {{{v[3]}}}.


(<==)  Let {{{v[1]}}} be a linear combination of  {{{v[2]}}} and {{{v[3]}}}.

==> {{{v[1] = alpha*v[2] + beta*v[3]}}}  for some real {{{alpha}}}, {{{beta}}}.

<==> {{{v[1] = alpha*v[2] + beta*(3*v[2])}}} , by hypothesis.

==>  {{{v[1] = (alpha + 3beta)*v[2]}}}, and therefore {{{v[1] = a*v[2]}}} for {{{a = (alpha + 3beta)}}}


The statement is thus proved.