If we have linearly independent vectors $v_1, v_2, ..., v_n$ and create a new collection of vectors $v_1', v_2',...,v_n'$ such that each $v_i'$ is a linear combination of $v_1, v_2, ..., v_n$.
Are vectors $v_1', v_2',...,v_n'$ linearly independent as well? I believe the answer is no, at least in general. How can one prove it easily?
Because $v_1, v_2, ..., v_n$ are independent, the equation $a_1v_1 + a_2 v_2 + ... + a_nv_n = 0$ is satisfied only if all scalars $a_i$ are zero.
If we wrote a similar equation for $v_1'$, namely $b_1v_1' + b_2 v_2' + ... + b_nv_n' = 0 $, after expressing each $v_i'$ as a linear combination of $v$ vectors, the new coefficients would have to be zero. Therefore the original coefficients $b_i$ don't have to be all zeros.