Find a plane of $R^3$ that doesn't contains any of the coordinate vectors.

Question

By giving a basis, describe a two-dimensional subspace of $R^3$ that contains none of the coordinate vectors $(1,0,0), (0,1,0), (0,0,1)$.

Seems not to exist. For all choices of planes (through origin) always will be some free coordinate who them could spawn its coordinate vector.

There is a solution?

Answer 1

Take the plane spanned by $(1,1,0)$ and $(0,1,1)$.

Answer 2

Why not try $U =\operatorname{span}((1,0,1),(0,1,1))$ ? Try Thinking Geometrically