Question

Either prove that this statement is always true, or give a counterexample to show that it may be false: If {v1,v2,...,vp} is a linearly dependent set of vectors in R^n, and x is any vector in R^n, then {v1,v2,...,vp,x} must also be linearly dependent.

Answer 1

If first system of vectors is lin. dep. in R^n then there are some system of
scalars {a1,a2,...,ap}, not all of them are zero,
that

a1*v1+...+ap*vp=0

If we consider system of scalars
{a1,a2,...,ap, 0}, then system {v1,v2,...,vp,x} will be also lin. dependent, in
the same way as before.