Suppose that {v1, v2, v3, v4} is a basis for a subspace V of

. Which, if any, of the following subsets are bases for V?

1) S1 = {v1 + v2, v2 + v3, v3 + v4, v4 + v1}

2) S2 = {v1, v1 + v2, v1 + v2 + v3, v1 + v2 + v3 + v4}

3) S3 = {v1 - v2, v2 - v3, v3 - v4, v4 - v1}

4) S4 = {v1 + v2, v2+ v3, v3 + v4}

5) S5 = {v1, v1 + v2, v2 + v3, v3 + v4, v1 + v2 + v3 + v4}

Justify your answers?

Can someone just show me how to do one of them, then I wil be able to do rest of them?

I know a basis needs to be a span of V and linearly independent, I am not sure how to prove it is a span of V