Question 460760
Let S = {   {{{v[1]}}}, {{{v[2]}}}, {{{v[3]}}}    } be a linearly dependent set of vectors in V.

This means that there are scalars {{{c[1]}}}, {{{c[2]}}}, and {{{c[3]}}}, not all of them zero, such that {{{c[1]*v[1] + c[2]*v[2] + c[3]*v[3] = theta}}}.  But then, we would also have {{{c[1]*v[1] + c[2]*v[2] + c[3]*v[3] + 0*v[4] = theta}}}, letting {{{c[4] = 0}}}; and not all of {{{c[1]}}}, {{{c[2]}}}, {{{c[3]}}}, and {{{c[4]}}} are equal to zero.  Hence {   {{{v[1]}}}, {{{v[2]}}}, {{{v[3]}}}   , {{{v[4]}}} }  is also a linearly dependent set.