Linearly independent set question

Question

If the vectors: $$ \left\{ (V_1,V_2 ,\ldots, V_n) \right\} $$ are L.I, prove that the same thing applies to: $$ \left\{ (V_1,V_2-V_1 ,\ldots, V_n-V_1) \right\} $$ What would be your way to prove it?

Answer 1

Assume there is a linear combination of these vectors with not all-null coefficients equal to $0$

Find out that it implies the original set of vectors is not linearly independant. This should be quite fast, since you have a linear combination of the original vectors, but it is the hardest part.

Conclude

Answer 2

Suppose $$a_1 V_1 + a_2 (V_2 - V_1) + \dotsb + a_n (V_n - V_1) = 0$$ for some $a_1,\dotsc,a_n$. This amounts to $$(a_1 - a_2 - \dotsb - a_n) V_1 + a_2 V_2 + \dotsb + a_n V_n = 0.$$ Now you should be able to conclude.

Answer 3

The matrix made with the coordinates of the vectors in the second set with respect to the given basis is $$ \begin{bmatrix} 1 & -1 & -1 & \dots & -1 & -1\\ 0 & 1 & 0 & \dots & 0 & 0\\ 0 & 0 & 1 & \dots & 0 & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots\\ 0 & 0 & 0 & \dots & 1 & 0\\ 0 & 0 & 0 & \dots & 0 & 1 \\ \end{bmatrix} $$ which clearly has rank $n$.