Originally Posted by

**HallsofIvy** Someone corrected my interpretation of this question. I though it was asking for a basis for the "subspace" S but in fact it is asking for a basis for $\displaystyle R^4$ consisting of vectors in the set S.

Try this: the standard basis for $\displaystyle R^4$ is <1, 0, 0, 0>, <0, 1, 0, 0>, <0, 0, 1, 0>, and <0, 0, 0, 1> where the last three are not from set S. Okay, just change the first component: try <1, 0, 0, 0>, <1, 1, 0, 0>, <1, 0, 1, 0>, and <1, 0, 0, 1>. Is that a basis for $\displaystyle R^4$? There are four vectors in it; are they independent?