# Is the set of sum of Linearly independent equations are linearly independent with the equations?

If I have a set of linearly independent equations, E1, E2, ..., En. I am wondering if all the possible sums are linearly independent between them and between the original equations.

{E1, E2, .., En, E1+E2, E1+E3, E2+E3, ..., Ei+En} are linearly independent?

If yes, how can I prove it? Thanks.

Of course not (unless $n=1$). Since your set contains $E_1$, $E_2$ and $E_1+E_2$, it cannot be linearly independent, because$$1\times E_1+1\times E_2+(-1)\times(E_1+e_2)=0.$$